IMU CET 2027 Matrix and Determinant Part 12 | Homogeneous & Non-Homogeneous System | Cramer's Rule

[00:00:38]Hello Everyone Myself Abhijeet Singh and I Well Welcome You to the Amazing Platform of Heman Opta and the Amazing Class of Mathematics. Lots of respect to Guru ji.

[00:00:45]Blessings to you son, lots of joy to you. Thank you. Good afternoon sir. Vishal Very good afternoon. Kanhaiya Mishra. Very good afternoon. Good in traffic Shubham came to know that Sir would be in traffic. And I am here son, I have come. Shubham, yes I am here. Sat Sri Akal Shubham. And sir please go ahead. Yes, let's just start now.

[00:01:03]Good afternoon. Shubham Guddi Rani Harsh Virendra and Mithun Kumar Good afternoon ok son let's start.

[00:01:12]Okay, so look, today we will finish it, we have the last concept of determinant left and we will finish it and this will be finished today, it will be completed, this is the last class, this is the last class of the concept, okay, we will finish the entire concept today and children, you used to say one thing to me that sir, how can we talk to you, in what way can you connect with me, you used to say so regarding children connect, sir told me that sir, you should do one thing, create an ID on Instagram for the children so that you can connect with them directly.

[00:01:41]So I said that because I don't run anything in Facebook. I am completely disconnected. I always stay away from the social media. I do n't understand it, nor do I get time.

[00:01:50]You have to teach on YouTube, that's why I come here. Isn't it? And there is some office work on Facebook and WhatsApp, that's why.

[00:01:54]So I have made this especially for you.

[00:01:58]So whatever is there you can connect with me directly.

[00:02:00]And see what you have to search on that? You have to search ina on Abhijeet Abhijeet unders Maths. You will search this.

[00:02:11]Ok? I have to enter this in this manner because sir told me to enter this and search. Absolutely exactly, if you put this then you will be able to see me directly there. And look, the photo that I have posted here, the photo that has been posted there is also the same one. Where is my photo, do you see it? The photo you see on the thumbnail. Ok? Whatever photo you see on the thumbnail, you will see that same photo. Ok? So you will recognize me from that.

[00:02:32]Yes son, I made it. Oh that is done.

[00:02:34]You can connect with me there. Ok? So there you will definitely be able to connect with me in a great way. Okay, so son, the photo of mine that you see on the thumbnail is the same photo that you see on the channel.

[00:02:51]Yes, you can look there.

[00:02:53]I made it only because look, I am not very active.

[00:02:56]Children kept commenting that they wanted to connect with me, Sir. I want to talk to you directly.

[00:02:59]So I used to say that you will go to the website.

[00:03:01]There we will talk to ma'am or sir.

[00:03:02]He will make me talk to you. So it becomes a little lengthy process. So sir suggested me to do it this way. So if you search this, then whenever I get time, most probably I get time on Saturday, then whatever you want to ask, whatever you want to know, I will reply to that. Well, you have got the channel, that is a very good thing. Okay, then it's fine. Yes, look, let's go.

[00:03:24]Yes Shantanu, yes son, I just saw that I received a request from Shantanu.

[00:03:30]He followed it. Ok. So let's do it this way, ah yes, it's absolutely fine. So let's start the class quickly. So look, whenever I get time, son, I am very active on social media. Yes, okay son, whatever you say, look, this has given you a platform to connect with me directly.

[00:03:47]So, there you can connect directly, talk to me regarding your class, regarding the content that we are studying, but we will talk about studies related to you, children. Okay, so now look, I am not always available on that, but I will be available, your message will reach me directly. It has happened like this, sir, I have a lot of pressure on my board, so that is why I will watch the recorded one. Yes, peace, keep watching the recorded one.

[00:04:07]No, no problem, watch the recorded one and these things will help you in your board as well.

[00:04:11]Ok? All these things are going to help you in the board also and in this way you can see. Ok? So whatever you want, let's see, you said it and we did it.

[00:04:19]So look at the way I followed my sir.

[00:04:26]Your following will increase. Yes sir, a sir, I have background who is intelligent, please call him. Good background that handles. Yes son, I am telling you that sir does not come at all.

[00:04:43]I had told this to Sir when we were starting the batch.

[00:04:45]I said come sir because look I am visible from the front here.

[00:04:49]But whatever meaningful things are happening in the background, Sir is taking complete care of it. We will introduce you someday, children.

[00:04:53]Ok? Yes, absolutely, I will introduce you to the sir. So I mean I'm working back here. So Sir, he gives so much support by sitting at the back end and everything keeps running smoothly. Isn't it? So, yes sir, this is enough of matrix. Yes, son, your lecture is a big topic and our topic is also very good.

[00:05:13]Let's go, our class is of 30 minutes each.

[00:05:15]Let's start quickly.

[00:05:17]Yes, okay, okay, Guddi Rani Vishal, okay, it is 67.3, it will be done. Yes, why not, it will be done. It will be done in all. More than this.

[00:05:24]In some companies, it is 60% enough. So, let's start our class.

[00:05:32]Okay, let's come and see, so see, today we will talk about the method to find the number of solutions.

[00:05:39]Okay, how do we find the number of solutions? And one more follower must have come, Harsh. Yes, of course, I know, it must be Harsh. Of course, I know, it is Harsh. Let's see, the method to find the number of solutions is homogeneous system. So, look, the first thing we will do now is to tell you, son, the equation. Yes, okay, okay, it is not there.

[00:05:58]Thank you. Look at them one by one. We will do it only if you have something important to say. We will see there. Ok?

[00:06:04]Let's go then. So method to find number of solutions homogeneous system. Now we will see, look this is the equation.

[00:06:10]This is an equation. The second equation is this.

[00:06:11]This is the third equation. Well, if I write this equation like this once here, if I write this equation once here.

[00:06:17]Like a1x + b1y + c1z = d1. There will be some constant term like this. Yes, okay. Ok.

[00:06:28]Now let's take a look at it carefully. Now let's study, this will become a2x + b2y + c2z = d2, so the third equation is a3x + b3y + c3z = d3, so these are the three equations I have, sir, there can be two equations, there can be two also.

[00:06:46]Okay, but we're going to talk about three equations.

[00:06:48]Okay, let's see if you listen carefully what is the difference between homogenous and non-homogeneous.

[00:06:55]Son, if all these constant terms become zero, then what does our system become like? Hey brother, our system has become a homogeneous system. Will it depend on this? Yes, it will definitely depend on whether the lines here, the constant here is zero or not. If there is zero in this way then it is a homogeneous system.

[00:07:22]Well, we are going to see what a question is made of from a homogeneous system.

[00:07:24]Sir I just joined MA Opta class.

[00:07:28]How can I view previous classes?

[00:07:30]Please. Son, the playlist is ready. If you go to the entire playlist, you will find all the classes. So then we saw here this zero this zero this zero. So what became of us? We got a homogeneous system of equations. I will tell you about non-homogeneous later. Let's look at Homogeneous first. What question comes and how can we solve it.

[00:07:44]Now see what we will do?

[00:07:48]What will you do once you have this homogeneous system of equations, my dear? Can I write this in a way?

[00:07:53]Look, I will show you now. Can I write this down? Look like this, I am writing it again.

[00:07:56]a1 Hey change the color. Ok. Let's change the color.

[00:08:01]This a1x + b1y + c1z = 0 a2x + b2y + c2z = 0 a3x b3y + c3z = 0 Now see I taught you matrix multiplication so can I write it like this a1 b1 c1 a2 b2 c2 a3 b3 c3 of course I can write it like this and here I can write x y z and here I can write the null matrix will become a null matrix. Correct?

[00:08:42]Because if you solve this, this is what you are going to get.

[00:08:44]So this matrix that I got, if I accept this matrix, I can write it like this.

[00:08:49]I considered this a. I considered this a capital x.

[00:08:52]So I can write ax = null matrix. This way I can write.

[00:08:57]So look, my dear, the important thing is what I am going to tell you here. The important thing is that we have to see the determinant of this matrix.

[00:09:05]Correct? If we have its determinant, sir, if it is not zero then son, if it is not zero then it will become non-homogeneous. Non- homogeneous After this I will get it done for you.

[00:09:17]Ok? Ok? Ok? So now we have to see whether this is zero or not zero.

[00:09:24]We have to look at these things.

[00:09:27]Correct? So, first of all, see, you will not do this much that you will keep writing like this again and again.

[00:09:31]A matrix made up of simple coefficients, we have to see whether its determinant will be zero or not. Let me show you now. See how to do it. Now look here, if that zero equals two does n't come. Ok? If zero equals two does not come then will you understand what it means? Look, now understand carefully. We will understand this very carefully.

[00:09:52]too much. I am telling you a very important thing. If you remember it once, it will be a lot of fun.

[00:09:58]Correct? So you just have to understand this thing carefully. So, yes, look, if the determinant does n't equal two to this zero, then what? So this is unique solution or x y z = 0 or trivial solution or sometimes it can also be said that zero solution.

[00:10:27]So whenever all these statements and this which is written here determinant is not equal to 0, it is the same thing. If ever he asks you what kind of solution is this, a unique solution, then what will come to your mind is determinant is not equal to zero.

[00:10:40]If he ever says x = y = z = 0 the only solution that will come is zero.

[00:10:47]So the determinant is not equal to 0. If he says tribal solution then this should come to mind. The determinant is not equal to 0. Even if he says there is zero solution, this thought should come to your mind. And if this is ever written then these things should come to your mind.

[00:11:01]I told you how we think in questions.

[00:11:05]This is what I want to tell you. Ok? So let's move ahead. Now, sir, if it comes equal to zero then I will tell you what it is? So what will happen?

[00:11:15]How will the system be solved? Infinite Solutions My dear Infinite Solutions. Ok? Or what shall we call it? Non- trivial solution. What can we say?

[00:11:26]Non-trivial solution. Non-trivial solution.

[00:11:33]Ok? So, these are the things you have to remember.

[00:11:36]Okay, look, you can remember it this way also.

[00:11:39]If the determinant is not zero, there will be zero solutions. 3 / 0 has the same meaning. So, it's the opposite, right?

[00:11:45]I am telling you the method of remembering in this way.

[00:11:46]If the determinant is how much is coming? The determinant is not equal to zero, so the solution is zero. And if zero solution comes, then it means everyone's value becomes zero.

[00:11:55]So this is what is called unique. Hey, unique means that there is only one person like him and no one else. Just a unique one, as unique as you are. Ok? And he says if the determinant equals zero, what will happen here? There will be non-zero solutions here.

[00:12:07]Non-zero means there will be an infinite solution.

[00:12:08]So you have to remember this also.

[00:12:11]Yes, Infinite Solution Harsh absolutely right. You are absolutely right. Okay, let's move ahead then. Let us look at some questions on this.

[00:12:17]Look, we have found the number of solutions. Tell me the answer quickly.

[00:12:20]Who will tell? quickly. Hurry up.

[00:12:23]Tell me the answer quickly, what will come?

[00:12:25]Start solving it very quickly, we will start solving it very quickly, let's solve it very quickly, children, we will start solving it very quickly, let's solve it very quickly, tell me what answer is coming, we will read the question and answer it, everyone has to answer it after reading the question, okay. Look, he is saying find the number of solutions. Want to know the number of solutions.

[00:12:58]Okay, that's good afternoon Rohit. I'm absolutely absolutely fine.

[00:13:02]Rohit, how are you? Will solve the question.

[00:13:06]So look find the number of solutions. Number of Solutions wants to know from us that brother, how many solutions will it have?

[00:13:12]Well, is he trying to know the value of x y z? No. He is not interested in this thing.

[00:13:17]First of all understand the question. He is not at all interested in what is the value of XY and Z? He is interested in this thing that what will be the number of solutions brother, solution zero, there may be no solution, there may be one solution, there may be two solutions or infinite solutions, he is interested in this thing, I am not seeing your comments children, comment quickly, I want the answer, comment quickly, Arjun probably did not come today, let's go right now, Arjun Shahnawaz both are absent, let's go right now, no problem, quickly Harsh Rohit Virendra Harsh son, reply quickly, solve it quickly, look at the question, what method did I tell you, find the number of solutions, then make the determinant from the coefficient. I know it will take some time to solve it but still I keep pushing you so that you do it quickly. So I create here this determinant of x is the coefficient one. Ok? The coefficient of x is one and this is 4 1 4 - 1 and this is 3 - 4 and -1 3 - 4 -1 1 -3 and then +1 will check once again.

[00:14:19]1 4 - 1 1 - 4 - 1 3 Yes 1 - 3 1 - 3 1 Okay so I got this.

[00:14:28]What should I do now? Its determinant value has to be found. This is my a determinant value of a. Now I want to solve this and find its determinant value.

[00:14:34]So let me see, first of all I want to make things a little easier by applying some row or column operation to it.

[00:14:40]So I do this work here.

[00:14:42]Well what if I'm here? If I add this to this column, it will become zero.

[00:14:48]Well, subtract four times this from this, so this one will also become zero, and mostly we want to make zero. D option has come. Infinite solution. Well then I thought I should talk to you a little.

[00:15:01]Yes, what are you saying son? Everything is good. I was watching sir's class. Okay, great, very good. Yes. Keep watching the class. If there is any doubt, we will discuss it. D, okay, I don't think D will come. Yes B Harsh, yes yes absolutely Harsh, absolutely I am finding Harsh's answer correct B absolutely right son there is no need to comment so much, just tell me once, okay Harsh, I am finding your answer correct, so let's see what I do, you can open the determinant like this, there are many ways, but if you apply row column operation, the calculation becomes a little easier and we should try how we can create maximum number of zeros, so what is going on in my mind, I am thinking that I will have to expand it according to any row or column, so if the element of the row or column becomes zero at the most, then I can tell the answer. For example, if I expand this row and two of its elements, one element or all three become zero, then I can quickly tell the answer. So what I mean is that I have to make as many zeros as possible. So I thought in my mind that I will expand it according to the first row.

[00:15:59]So I'll see if I can create zeros in how many elements of the first row. So I saw I could change C2. How? C2 - 4C1 OK?

[00:16:10]Due to which this will become my zero. Do n't let me take the rest, we can keep becoming anything.

[00:16:13]Well, what will this be? How can I change C3? C3 + C1 I can pay here. So let's do it.

[00:16:18]Well, look, there will be a change between two and three. There will be no change in the forest.

[00:16:22]So let's write forest like this. 1 3 And what will come here? The forest has arrived. I wrote forest. Ok? What should I do now?

[00:16:30]Four times this has to be subtracted from this.

[00:16:31]So 4 - 4 * 1 will become 0.

[00:16:35]We have to subtract four times this from this.

[00:16:37]So how much will -4 become? Minus 12 will become minus 16. That's okay. Well, how much is there here? 4 * 3 = 12 - what is k 12 minus k 1? minus k will become 13. Correct? Because from this I am subtracting it four times.

[00:16:51]Ok? Now come to this part. What do we have to do here? Just add column one to column three.

[00:16:55]So three added to zero-1 is two. One added two to one. very good.

[00:17:01]So this is what we saw here.

[00:17:03]Okay, let's move ahead. Now see, now it has become easy to expand.

[00:17:07]If you do it according to the first row and expand it according to row one, then only one will come here and what will be left here -16 2 and how much will come here, here will come -13 and here will come 2, okay, so multiply it a little, then how much will it come minus 32 minus minus what will it become, 13 2 = 26, so this will come to me -32 and this will become plus 26, so see what is left, 6 and 16, see is 16 coming? Minus 16 is coming. I think. Ok? No no, it is not minus 16. Ah yes this will happen, six is ​​coming children.

[00:17:49]Subtract this for 1 second. If we subtract 26 from 32, then 6 out of 12 becomes six. And here two will be left. So this will come minus six is ​​coming. See, is it coming to minus six only? How much is coming? 18 is coming. Where did it go wrong then? Did I make a mistake in the calculation?

[00:18:04]Your 18th is coming. Option B: Yes, option B would be correct. But how much determination, my dear determination, are you getting? Please tell me. Let me check it once again.

[00:18:12]What did I do? 1 4 -1 ok 3 - 4 and -1 1 -3 and 1 1 -3 and 1 what did I do? In column two, subtract four times column one from column two.

[00:18:22]So if I do -4 -12, how much will it be? Minus k 16 ok and -3 and -4 so this will become -7 ok this is wrong here, right, this is wrong my dear here -4 - 1 so - k 7 will come here ok here we just added 0 2 and 2 let's solve this quickly what will I put in place of -13 here I will put - k 7 here so how much will this become let's see now ok let's see this quickly this will become minus k 32 and how much will the minus become minus k 14 so this is -32 + 14 so now we will get minus k 18 ok so tell me what is the determinant coming oh brother the determinant is not equal to zero so what will be the solution brother will it be trivial will it be zero or will it be unique ok? All these three things will be correct and I had told you the meaning of unique, unique means one solution. So only one solution will come. So look, it was easy to frame the question.

[00:19:34]Ok? I will get it done son, I will get everything done.

[00:19:36]Son, you have come for the first time today.

[00:19:38]You didn't even tell me your name. Isn't it? And you are spamming. You keep messaging me so many times.

[00:19:44]Correct? So let's go, that's what we are.

[00:19:48]Well then. -It will be 18 only. Absolutely awesome. Look, this is confidence.

[00:19:52]Kanhaiya said Sir it will be 18 only. Look, this confidence should come within you children.

[00:19:56]Yes -18 -6 -6 Yes -18 will come my dear that is so see the main thing we said was if the determinant is not equal to zero then what will be the solution? The unique solution will be ours. So let us move ahead.

[00:20:12]We will now look at the next question. Okay, yes, look, this has a non-tribal solution. Now look and understand what this is about. I said when he says non- trivial solution. Non-trival solution means non-zero solution and we have read that if the reverse is non-zero then how much will the determinant come, will it come zero, okay Akshat son, don't comment so many times, you are spamming, do n't do this, study well, yes, there will be an exam in this interview of sponsorship exam, sir, there will be an exam for sponsorship or in the interview, son, there are exams too, there are exams for that, first there is an exam, I will teach you, I will prepare you for the written exam for that also, come and see, now he said, he said trival solution and trival means non- tribal, right, yes, non-tribal means non-zero solution, and what is the determinant in non-zero solution? The opposite happens.

[00:20:59]Meaning if it says zero then it will be determinant non zero. If zero speaks then it is non-zero and if non-zero speaks then it is zero. So here he said non trouble. Non-tribal means non- zero. I mean, what will be the determination, my dear? Our determination will be zero.

[00:21:09]I just told you these things.

[00:21:11]Look at this, look at this. I had written it here.

[00:21:15]Where did you go? Here it is. Let's see. He said non-trivial solution. Non-trivial solution means determinant zero. When the solution is zero, we keep not equal to zero.

[00:21:24]Non-tribal means non-zero solution. It also means non-tribal, it also means non-zero.

[00:21:27]So the determinant equals to zero is just the reverse. The determinant is the opposite of what he tells us there. So in this way I am also telling you how you can remember these things.

[00:21:35]Let's go then. So just what do we have to do?

[00:21:37]Remove the determinant. Put equals to 0.

[00:21:39]And you'll get the value of a what he wants to know lambda. Ok? So first of all we will make three and -2 and solve it quickly children. We will keep solving it together.

[00:21:47]Your pen should not stop. Come on, how much is it?

[00:21:50]How much is 3 -2 and 1 here? Lambda okay how much is it here? 14 to the minus, okay, how much is that right here? 15 So lambda -14 and 15 1 2 and -3 over here is 1 2 and -3 okay? So what should this determinant value be?

[00:22:08]Your determinant value should be zero. So now we have to solve this.

[00:22:12]Or apply some operation on rows and columns etc. and see how you can solve it. Ok? Or one way is to expand it, we can solve it in this way also. So if I look here, if I look at one thing, if I subtract three times this from this and add two times this to this. Okay, let me do it like this.

[00:22:32]Because I want to see what's going on in my mind. Which way am I thinking? I am thinking of doing extensions according to the first row.

[00:22:38]So according to the first row, if any element becomes zero then that part becomes zero. If three parts come then it becomes zero. So I try to ensure that at least two becomes zero. So it's easy. And when is it easy to become zero?

[00:22:49]When you find even one element.

[00:22:51]You guys will remember this. Ok? No channel form. So you will remember this thing.

[00:23:00]Yes, I am not seeing the comment.

[00:23:02]Actually, let's look at the screen.

[00:23:07]So yes, look, I think in this way that if even one element becomes one, if even one element becomes one, my dear, then what can we do, we can make the remaining two elements zero. If you look at me and see one sitting here, then I can make this also zero. I can make this also zero. Why?

[00:23:22]Because if I multiply it by one and multiply it by three, I will make three and subtract it and I will get zero. If I multiply one by two, it becomes two and if I add it to minus two, it becomes zero.

[00:23:30]So this is my way of thinking.

[00:23:32]You too must have thought in this way. So the question is approached like this. So what did I do?

[00:23:37]I do that C1 is replaced by C1 a - how much? 3C1 Okay. After that C2 is replaced by C2 + 2 times C3 this is the way I am doing it. So look, it will go through this.

[00:23:54]What will happen when we subtract three times from three? This will become zero. Ok? This is zero. Well, if we subtract three times this from lambda, then what is three times 15? 45 So my dear this is Lambda - 45 Okay what are we doing with this? If we are subtracting three times, then three times this will be -9 and when we subtract -9, it will become +9 and this will become 10, so I have 10. Well, what to do after that? Then double it. If you add this, it will become zero. Ok? Then double it. How much will this be? 13. If you add 13 to 14, what will it become? It will become 16. Ok?

[00:24:31]How much do we have to double this? He is doubling it, right?

[00:24:32]Yes, double. Do six six and add two to it, what will be -6 and +2? - will become 4.

[00:24:38]So this becomes -4. This change will not happen because the change is happening in C1 and C2. There is no change happening in C3. There is no change in C3. So this is going to be 1 and 15 and this is going to be -3. Is it okay my dear? So now see, expand it accordingly. Now it has become easy. Plus minus plus and how much will this be?

[00:24:58]Lambda - 45 and this is 10 and this is 16 and this is -4. So we can expand this. Okay, four can also come out as common from here and it will remain equal to zero.

[00:25:10]So four if I take the common from here then it will become 4 inside lambda will be left - 45 and here there is 10 and here 4 and here -1 = 0 and further if we solve it further then I see here this will become - lambda + 45 okay and this will become minus k 40 = 0 I will have it okay so this will come to minus k lambda and this will come to minus k plus k 5 = 0 now tell me where is zero son, not zero where did four go? Four went into zero.

[00:25:42]Otherwise, the value of lambda will come to me as five. So I think your answer must have been five.

[00:25:49]Yes, tell me quickly.

[00:25:56]Now son, see, this is the way. This is the way.

[00:25:58]This is the way deterrent equals to 0 and what are we doing? We are only solving the determinant. We just solved the determinant. And what did we do? Isn't it you?

[00:26:09]Yes, okay. So this is yes five is coming.

[00:26:13]very nice. Very nice kids.

[00:26:15]very nice. Ok. A lot of fives coming up.

[00:26:17]Very true. Your answer is very correct.

[00:26:19]Yes. The value of lambda is five. Kanhaiya's answer had come long ago. Five is our answer. Ok? So in this way we can solve the question. But you have to solve the determinant and get the answer. Ok? This thing has been added and it does not take us any time.

[00:26:30]We just have to solve the determinant and the value of the determinant will come to us. Yes, good afternoon Ganesha. Good afternoon. Harsh also got answer five.

[00:26:37]very nice. Very nice kids.

[00:26:38]You guys have given a very correct answer. So look, you just have to remember these things. The question that will come in your exam will come in the same manner as I showed you this question about lambda. This is the question that I told you. Now see, this can happen with what we have and what can we do about it? If you think about it in a different way, then you can think about the question in this way also, look, if I have this kept with me here. Isn't it? This is the value I have kept here.

[00:27:02]If I put five value here.

[00:27:04]I put five values ​​of lambda here. Or like look here, if you want to try it with the lambda option, then if I put the value five here with the option, then you should see this thing, we end the question right here, we settle it right here, how son, try putting the value here, if suppose we put the value five here, then you will have to feel this, if you can feel it, look at this thing, this is three, this is five and this is one, this is the first column, well, look at this column, this is one. This is 15 and this is -3. Are you able to feel that this is our 15. This is -3. K -3 Times Yes, something is wrong here kids. This will not happen like this. No, this will not be possible. Because what was I saying? If I can see the proportionality.

[00:27:49]If I see proportionality then the determinant becomes zero.

[00:27:52]This is what I wanted to say. Because if I look at this, it is not three times or five times.

[00:27:57]I don't see the proportionality here. So we can't do it. Or the second way children is that you solve it by putting the values ​​and checking them. But I say that this is our best way. All we have to do is solve the determinant. Keep it equal to 0 and get the answer. Ok? Yes. The question has to be solved in this manner. This is how we solve it. How are you subtracting the column, son? I told you this.

[00:28:16]We had this property. I studied the property of determinant. I had said that this operation is subcent.

[00:28:21]Whenever we have to make zero. When a column is subtracted from a column, each element is being subtracted. If I'm subtracting this three times from C1, right?

[00:28:28]Like here I subtracted. Subtracted it three times from column one.

[00:28:31]So three times this is subtracted from this.

[00:28:35]Three times of this was subtracted from it.

[00:28:36]Three times of this was subtracted from it.

[00:28:37]Each and every element gets subtracted. Because I did the entire column three times. And I am subtracting from this entire column. So element to element is subtracted.

[00:28:46]I told you this. Ok? Yes. So let's move ahead now. Let's go quickly.

[00:28:52]Now let's see, now what we have is what we have now what you see is that we have non-homogeneous equation. Now let me tell you what type of questions we have from non-homogeneous.

[00:29:04]Son, you should practice this once. Virendra, you should practice this once. This PDF will be uploaded to the group now. You should try this and answer the questions again. Try it once.

[00:29:13]Yes, I told you in the previous lecture.

[00:29:16]Harsh said absolutely. I had explained this in the previous lecture. It is simply a matter of determination. Yes, look at it, Kanhaiya is saying it simply.

[00:29:24]Solving a very simple determinant.

[00:29:26]This means children, you did not see this previous class. Those children who are having doubts should definitely watch the previous class and in that you will know why we had told about the property. Isn't it? And then let's move ahead. The next type that comes to us. Isn't it? Today in today's class, I have to tell the children very simple things. The remaining determinants have to be solved.

[00:29:43]We have already read how the determinant is solved. Isn't it?

[00:29:46]How are multiplications or operations performed? I had told him about all the property. This is the work there. Here we have to tell you some more things about how to solve equations. Look, understand one thing carefully. This is its last topic and it is very useful. If the constant terms are not zero. If our constant terms are not zero then what will become of us? That will become our non-homogeneous system. Ok? What kind of system will be created? That will become our non-homogeneous system. Yes. Yes, no problem son, you will get it done if you practice. Shubham, don't worry, it will get done if you practice. You will have to answer the questions many times. Rest I will make you revise and practice the questions there.

[00:30:23]Okay, now see, this d which is non- zero here, so what will it become, this is our non-homogeneous system, and see, the most important thing in the non-homogeneous system is d, so what is it, we have just kept its coefficient, keep listening carefully, I picked up this coefficient and kept it here, so I got d. How to get a good d1? Look, D1 will be found like this, from the first column one, remember one means the first column is D1, so the first column has to be changed and the change has to be done by the constant term, okay, the first column which was D1 D2 D3 came in place of the first column, okay if I do D2 two means there will be a change in the second column, that is, what will we do in column two, instead of column two we will do D1 D2 D3, okay, then D3 three means the third column will change, that means D1 D2 D3 will come in place of C1 C2 C3, so we have to remember these three things.

[00:31:12]This was an important thing for us as to how to place d, d1, d2 and d3. If you set this sir, you will get 7 PCM yes. I will definitely get it.

[00:31:22]You just have to clear the sponsorship exam. I will definitely get it. Yes, I will get it. So let us see this. Okay, look at this first task, there are two-three steps, we will remember them.

[00:31:30]I told you how to write it this way.

[00:31:31]Ok. What next? Next up is this. Now look, understand this thing. We read this thing. You can read it in the book also but we keep forgetting it. The problem is not that I do n't understand. It is understood that the problem is of forgetting. We keep forgetting this. Yes, I will get it done tomorrow. We keep forgetting this.

[00:31:47]This is a big problem. So look, I am telling you that now. We will understand this in 2 minutes of peace.

[00:31:51]How am I teaching? You will remember it and will never forget it.

[00:31:56]Ok? Let's see now. Look if our d is non-zero. What we just figured out is what I told you. This one d, this one which is d. Correct? This if non zero is talking about this d. When these people have no contribution in this. Just the coefficient is their contribution to our variable.

[00:32:15]Okay, let me write one more thing here.

[00:32:16]Look, the value of x that comes here is d1 / d, sorry d1 / d, the value of y is d2 / d and the value of z is d3 / d, okay tell me one thing, determinant, I told you what is determinant? A determinant is a value. I told you from day one that deterrent is nothing but the value.

[00:32:43]What is a matrix? The Matrix Is Nothing But The Arrangement. A matrix is ​​an arrangement.

[00:32:47]Determination is a value.

[00:32:48]I have been telling you this since day one.

[00:32:51]Why should I take admission in PCM college? One should take these drops and prepare for university.

[00:32:55]Please tell me sir.

[00:32:57]How much in PCM? 65.7% admission is required. Son, prepare well and take a good college where you can get placement. This should be your target. So, I told you that d is a value. When I remove x and let d0 be the solution, will I get it? Will not get it. So just remember this a little bit. I will tell you further now.

[00:33:15]I just want to tell you this. d is a value that we get from here. If this is zero then we will not get a solution. Now see what I am trying to say. Let's look at the first statement. If d is not equal to zero then everything is fine. System Has Unique Solution. Everything is great right away.

[00:33:29]Look, this is a unique solution. Because I just told you. Hey brother, if this becomes zero then where will we find the solution? If it is not zero then we will get the solution. How to get it? Look here. Look at this. So how will you find the solution? You should check this first. d is not equal 0 That's okay. System Has Unique Solution. I will solve it quickly. Will remove it.

[00:33:46]This will come. One solution is this, second solution is this and third solution is this. That's okay.

[00:33:50]Everything is great. The matter is completely set. Correct? Well, when did the problem occur next?

[00:33:55]Now remember it this way. I am telling you the way to remember it. We will remember in the way I am telling you.

[00:34:01]see how? You will say that D is the leader of all these.

[00:34:04]D1, D2, D3 are the leaders of all.

[00:34:07]He said let's all become zero together.

[00:34:09]He said leader let's protest.

[00:34:12]All of them together become zero. Everyone said, okay boss, we are also zero. If you are zero then we are also zero. Everyone listened to what he said and everyone became zero. And you know unity is strength. There is power in the organization. Isn't it? It is said that unity is power. And when they got power, they got infinite power and our solutions became infinite.

[00:34:29]We have infinite solutions. Correct? Don't read this. Just look at the infinite solution.

[00:34:34]Ok? How many solutions do we have? We have infinite solutions.

[00:34:38]Ok? Good. Now again it happened that he said now what could be the case that the leader would say that let us all become zero. But even one of these three became a rebel. He said no, I will not agree. If even one person said this, the group would break up. Then say one, say two or say all three. Ok? So he said then system has at least one off means at least one or D1 should say that I am not zero.

[00:35:04]I don't know who you are, you biter. I will not listen to you at all. Ok?

[00:35:07]Or D2 says I won't agree. Or say D3 or say any two of these or say all three. There should be at least one person who refuses to obey the leader. And as soon as he agreed, the group broke up, the protest failed, and the job was over. Ok? So what will we say? No solution. Hey, will any solution come out then? When there is a split in the group itself, it will not be available. Ok? So I told you this method of remembering. This method because I know that there is a problem in remembering it. Ok? Everything will become zero. Yes, everyone said yes, let everything become zero, I have found it.

[00:35:37]Yes, that's exactly what the leader said. So, you have to remember it this way.

[00:35:41]I told you how to remember this. Ok?

[00:35:43]Come on now, I hope you won't forget today at all.

[00:35:46]Now you are the way I am telling you. Let's go quickly. Find the number of solutions this this and this. Now he wants to ask you the number of solutions. So what will you do? First you will find out d.

[00:35:55]D What will happen to us? 2 - 1 and 1 well how much will it be? 1 3 and 2 and how much will this be? I wrote 3 2 3 3 2 3. I wrote 1 3 2 and here I wrote 2 - 1 and 1 okay? Now see how to find its determinant.

[00:36:11]First we will apply row column operation.

[00:36:13]So what do I do? First of all, take a look.

[00:36:15]Look here, how I am doing. I am doing c2 is replaced by C2 + C3.

[00:36:20]Correct? And what do I do? Ah C1 is replaced by C1 - 2C3 in this way the operation I am applying is if we subtract two times this from this then it will become zero. If twice this is added to this, it will become 1 - 4 - 3. Well, twice that amount will go into this. How much will 3 - 6 be? 3 - 6 - 6. Okay, three and its double 3 - 6 will become minus 3, okay. This became -3. Okay then I said add these two. Column two means add these two columns. Look son, look at the one who was saying that there is confusion. So both of these have to be added to it. There will be a change in this. I have explained these things in great detail.

[00:36:53]You should go and see the previous class once.

[00:36:55]So we are adding this to it. So 1 - 1 and 1 0 2 and three how much do we have here? 2 and 3 became 5 and three and 2 became 5. Correct? Well, there is no change in this here. The change is happening in c2 and c1.

[00:37:06]1 2 3 Look son, you will feel it here that the determinant value is coming out to be zero. How? I told you that if any two rows become proportional or equal then what will be the answer? Zero comes. Now you see for yourself. Or you can feel this right here. If you are not able to feel it then I will make you feel it.

[00:37:22]-Took 3 commons, took five commons. 0 1 1 0 1 1 2 3 Two rows are the same. The two columns became identical. How much did you get as an answer? Zero. Determinant zero. Hey, what kind of case is this, sir? This case has become that it is not zero. d = 0. Now either this can happen or this can happen. Now either all three of these should become zero by accepting the matter of litre. When he said let's become zero together, they all agreed.

[00:37:46]So how many will there be? Infinite solution or maybe even one of them will come forward and will not listen to the leader and will refuse saying who are you? We don't agree with you.

[00:37:53]Ok? So now let's check d1. Let's quickly check d1. d1 means the first column will be replaced by the constant term. This will become 4 12 and this will become 10. Okay, 4 12 10, everything else will remain the same. What is a bean? -1 3 2 1 2 and three this will be it. Now when you solve this, son, this zero will not come. You solve this. It's class time so I am not solving it. When you solve this, this will not equal to zero. So we just needed one.

[00:38:24]We needed just one person who did not listen to fear and instead listened to what was there and what happened to us?

[00:38:32]became non-zero. Bad work, no solution.

[00:38:34]Harsh absolutely great. Harsh's answer was coming out absolutely correct. Ok. Yes very nice. Come on Shubham. Infinite solution. No, Shubham Infinite will not come. No not infinite.

[00:38:46]How will we get these solutions? d1 is not equal to zero. Ok? So no solution. I told you there is no solution. Okay, let me give you one more question in your homework. If you remove its value, it will come out non-zero.

[00:38:55]Ok? Let's move ahead. Now look, there is a solution here. I will just change it a little bit.

[00:38:59]Ah d2 / d this is how much language is written in this question. There is nothing in this question. Brother, what does he want to know from us? What is the value of d2? d2 wants to know the value. Do we know what y is? d2/d.

[00:39:13]First he told us, see children, what happens? This equation is given to us. So when we get the value of y, we get it by dividing d2 by d.

[00:39:19]And we know why are you telling us brother? So we know about this.

[00:39:21]And he also said that look, this happens. Then I would have told the answer myself. Isn't it? That's all I have left. I should have told you the answer.

[00:39:28]So look what he asked us here?

[00:39:30]D2 has asked. And d2 what do we have to do? Column Two changes. 1 1 and one and what will come here on d2? Just how much will change? 7 16 and 7 16 and 22, okay, and this will remain the same, 1 3 and 4, son, solve it, its determinant value, as far as I think, if I am able to see it, then probably -3, its solution will come, once you solve it, this is our value d2, okay children, when you try to do this good work, many people get stomach ache and they do not let you do good work, these things will keep happening, we will teach in future also, a lot of people will create spam, they will come to our live class and will disturb you and me too, son, this keeps happening, okay?

[00:40:10]So just ignore people.

[00:40:12]Ignore people. Focus on your studies.

[00:40:14]I am also focusing on studies. And we will meet in the next class. Study well. Till then bye.