✨ KCET 2026 Maths 🔥 Matrices & Determinants Important Examples | Part 1#KCET#Matrices#Determinants

Added: 2026-05-26

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[00:01:01]First of all, we are beginning with the KCET examples on the two topics, matrices and determinants.

[00:01:10]If we talk about the weightage of these two topics is eight to nine marks. That is in the CET examination, they're asking eight to nine questions approximately.

[00:01:21]So, in today's video, we are discussing the examples KCET examples which they have asked for the year, that is April 2026.

[00:01:31]Beginning with the example number one.

[00:01:33]Here, the first question they have given, that is match the following list one and two.

[00:01:40]So, here first they have given, that is A is equal to there is a matrix A which is not a square matrix. As all of you know, that is a square matrix is nothing but in which the number of rows and the number of columns are same. If I take one example, suppose A, B, C, D. So, here we have two horizontal lines, that is these are called as the rows, and two vertical lines, these are called as the columns.

[00:02:08]So, this is a matrix with a order, that is two rows and two columns, and hence the order is written as that is two cross two. So, this is an example for square matrix. So, square matrix is nothing but in which the number of rows and the number of columns are same. So, whenever the rows and columns are not same, for example, if I take ABC and DEF, then here the we have that is two rows and three columns. So, it's a matrix of order two cross three two cross three. And if you see this, the shape of this matrix, so it it looks like a rectangle, and hence students, that is whenever we have number of rows and number of columns are not same, no, that whatever the rectangular arrangement what we say, it it looks like a geometrical figure that is a rectangle. And hence students, this matrix is called as a rectangular matrix. And this arrangement, whatever the matrix arrangement, this is what it looks like a square, and hence students, it's a square matrix. So, it the geometrical figure becomes a square only when we have a number of rows and number of columns same. Otherwise, it becomes that is a rectangle. And hence what we can say is a matrix which is not a square matrix means it is a rectangular matrix. So, A is with three, now you can write it here.

[00:03:28]>> [snorts] >> A is with three.

[00:03:31]Next, a square matrix A' is equal to A.

[00:03:34]As we know, A' is nothing but the transpose of the matrix. So, whenever A' and A both are same, what happens is if I take one example, suppose I'm having that is one two and two three.

[00:03:48]So, this is a matrix A. Now, we write A' here, That is transpose of the matrix and the transpose is written by interchanging the rows and columns. So, the first row, I'll write it as a first column. The second row, I'll write it as a second column. Okay, again what we got here A dash and A both are same. So, whenever A dash and A both are same, so whatever this matrix A is A is there, it looks like this. First of all, it is a square matrix, no doubt. And the second one is if I take this is our principal diagonal. So, if we take students that is this is a element A12 and this is A21. So, it is like a square. I said already it's a square matrix, so it is like a square paper. If I fold that is right to top to left bottom, then these two will coincide with these two. I mean A I J or A12 is same as A21. So, whenever A I J is equal to A J I, that matrix is called as symmetric matrix. I mean the corresponding symmetric pairs must be same. So, here these two, I mean A12 element is same as A21 and thus it is a symmetric matrix. So, remember whenever A dash is equal to A, it's a symmetric matrix and what we can say is B is what? One. So, that is for the second one.

[00:05:12]Beginning with the third, the diagonal elements of a diagonal matrix are same.

[00:05:18]So, first of all, it's a diagonal matrix they have told. In the diagonal matrix, the diagonal elements are same they told. First of all, students remember >> [clears throat] >> the diagonal matrix is nothing but in a square matrix in a square matrix, whatever the non-diagonal elements are there, diagonal elements you leave it, the remaining non-diagonal elements all are zero here. This is what we call it as a diagonal matrix. And what they said now, further, in a diagonal matrix, again, these diagonal elements are same, they have mentioned. I mean, I will take in general this as a K K K. Then we learned already in a diagonal matrix, whenever the diagonal elements are same, this matrix is called as students, that is a scalar matrix. Okay? So, C is with that is option four.

[00:06:11]Because for four they have given here scalar matrix. Now we have our fourth one. Okay, obviously this is the last one. So we left with two one four we have Sorry, three one four we have taken. So we left with the second you can mark directly. Uh okay, anyhow, what is that uh second one? So Sorry, fourth one we'll discuss. A matrix which is both symmetric and skew symmetric. Okay, they have said that there's a There is a matrix which is symmetric and as well as it's a skew symmetric, they say. So if it is symmetric, then A' is equal to A.

[00:06:44]And if it is skew symmetric, then the corresponding symmetric pairs are with opposite sign and hence you will get A' is equal to minus A here. Okay, this is also A' is equal to this is also A' is equal to. So what we can say is A is same as minus A. So this is matrix A.

[00:07:02]I'll bring this matrix also left hand side. So you will get here zero matrix.

[00:07:06]So A plus A, it's a 2A is equal to O, or you can say zero matrix. And the students from this you will get 1/2 into O means it is again O, and O is nothing but it's a zero matrix, or you can say it's a null matrix. So whenever A' is equal to A and A' is equal to minus A, you just solve this, you will get students the matrix A as a zero matrix here. And this what we can say is uh a matrix in which that is both If it is symmetric and skew symmetric, then definitely it's a null matrix. And thus we got here option A is with three, B is with one, C is with four, and D is with two. And that is nothing but option three.

[00:07:52]A three, B one, C four, D two. And that is option three. This is the first example.

[00:07:58]Now, let us begin with that is example number two.

[00:08:04]Consider the following statements. Okay, here they have given students some statements to us. Um Number one. A is a non-singular matrix, then A inverse exists. Yes, we know students, A is a non-singular matrix.

[00:08:20]For example, let us consider first what is A inverse means. So, inverse of the matrix is given by 1 / determinant of A into adjoint of A. So, in this what we have is in the denominator here determinant of A, okay? And they have mentioned in the statement one that A is a non-singular matrix. We know uh whenever the matrix is non-singular, determinant of A is always not equal to zero. And if this determinant of A is not equal to zero means obviously we never get this value as 1 / 0. I mean, we never getting this value as infinity.

[00:08:56]Then obviously there is no any problem to have that is A inverse. Thus what we can say, if the determinant of A is not equal to zero, then definitely A inverse exists. And hence students here the statement one is true. Okay? This is true.

[00:09:14]Now, second one what they said here, if A and B are symmetric matrices of the same order, then AB - BA is a skew-symmetric matrix. Okay. Uh if I take that is uh AB BA. Okay, to check whether it is symmetric, skew-symmetric, now we have to take its a transpose and we have to simplify. Let us apply the transpose properties. This can be written as AB whole dash minus BA whole dash. So, whenever you have a algebra subtraction, the transpose can be taken separately.

[00:09:50]Now, this is equal to further we learned AB whole dash can be written as B dash A dash minus BA whole dash can be written as A dash B dash. And again, what they have said in the statement is A and B are symmetric matrices. Okay, already A and B are symmetric means A dash is equal to A and B dash is equal to that is B. So, now we substitute B dash means B, then A dash means A minus A dash means A and B dash means B. Okay, we'll take here minus sign common, then this becomes students AB minus BA. And this what we can say here we have considered transpose and that is equal to we got minus of so we got minus sign here. This implies that AB minus BA is skew symmetric. And even in the statement they have mentioned the same. And this what we can say even the second statement is also true. So, here statement one is true and statement two is true and that is nothing but the option three. This is example number two.

[00:11:07]Next, let us begin with example number three. A row matrix has only Okay, here they have given the options as one element, one row with one or more columns, one column with one or more rows, and the fourth option they have given one row and one column. Okay, we learned already in the matrix students that is when we say that the matrix is a row matrix means Uh, we know already that matrix is nothing but it's a arrangement of the numbers you can take or the functions you can take now with some rows and the some columns. So, here a row matrix when we talk, so it has students row matrix is nothing but it has a single row. I mean only one row it has, okay? Only one row. So, we can write one example here 1 2 and 3. You see this example here we have exactly one horizontal line, okay? So, this is what we call it as a one row and we have students here that is three columns.

[00:12:11]Three columns. I mean three horizontal lines. So, this is what a example for row matrix. So, what we can say by seeing this example, it has exactly one row and for the columns we don't have any restriction here. This what we can say is row matrix is nothing but it has exactly one row with one or more number of columns. I told for the columns we don't have any restriction. It may have one column, two, three, four and so on.

[00:12:39]But the condition is it should have exactly one row. And this what we can say here the option is a row matrix has only that is one row with one or more columns and that is option two.

[00:12:52]This is example number three.

[00:12:56]Beginning with example number four. Here they have given students X and Z are the two matrices. Okay, first they have given here X is a matrix of order 2 cross N. And Z is a matrix of order that is 2 cross P.

[00:13:12]And they have told that N is same as that is P. Then the question is here, what is the order of the matrix that is 8X minus 9Z? Okay, if I talk students for the 8X, so 8 is a scalar here, X is a matrix. So, you know the theory. I Whenever you have that is an X is a matrix of order two cross n, then 8 into X when we do, even that is also a matrix. You have to multiply the scalar to the matrix. Again, you will get students a matrix with the same order that is two cross n. I mean this 8 X is a matrix of order two cross n.

[00:13:49]Similarly, 9 Z is also matrix of order same as of Z that is two cross P. Now, the subtraction is defined here. So, we learned the theory matrix subtraction. I mean A minus B is defined only when A and B are of the same order. Means obviously here two cross n must be same as that is two cross P. Then only the subtraction is defined. And what what we have one more theory is here the subtraction is defined when they are of same order. I mean A is a order of m cross n, B is a order of m cross n. Then after simplifying, I mean after subtraction, whatever the new matrix you will get, even that is also of the same order you will get that is m cross n.

[00:14:34]So, that's what we can say 8 X minus 9 Z. This matrix is either of order two cross n or two cross n is same as what we have got here two cross P. So, any one possibility is there among these two. And here they have given option that is two cross n. So, this is example number four.

[00:14:59]Now, let us begin with that is example number five. Which of the following is correct?

[00:15:05]So, the first option they have given determinant is a square matrix. Second one, determinant is a number associated to a matrix. Determinant is a unique number associated to a square matrix.

[00:15:14]And determinant is not defined for a square matrix. Okay. So, these all four points are for that is the determinant they have given means with respect to the determinant they have given these four statements. So, let us see students what is mean by determinant. So, first of all remember determinants are defined that is only for determinants are defined only for square matrices.

[00:15:43]This is the first thing.

[00:15:45]Determinants are defined only for a square matrices and determinant is nothing but it's a function. It's a function whose the domain is square matrices whose the domain is square matrices and whose codomain is that is set of real numbers. So, here f a determinant is a function. It is from square matrices to that is the set of real numbers it is defined. So, what they say is determinant is a real valued function. So, if I take one example, see here we have a domain. In this domain students we have a square matrices. So, we have to solve this square matrix. It may be of order one cross one, two cross two, three cross three and so on. Solve that square matrix determinant, you will get students its determinant value as a real number here. Okay? And that's why what they say determinant is a real valued function. So, whenever you solve any square matrix determinant, you people are getting that is a unique real value here and that's why what they say it is determinant is always a unique number.

[00:16:56]Okay? Whenever we solve, you take a one cross one matrix. Suppose they have given a matrix now that is two with a this is a matrix of order one cross one.

[00:17:05]So, it's a determinant if I take this as a A then determinant of A is two. Or suppose if you have here minus two, if it still it is a one cross one matrix, that time students the determinant of A is written as that is minus two. So determinant is defined for a square matrix first and whenever you simplify, you know, you will get a real value.

[00:17:27]That's what we can say determinant is a unique number associated to a square matrix and that is option three. This is example number five.

[00:17:38]Let us begin with the example number six that is if A and B are invertible matrices, then which one of the following is not correct? Okay, the first and second they have given that is A into adjoint of A. Here also we have A into adjoint of A. And the third one they have given AB inverse. Okay. Uh we know students AB inverse uh is always equal to that is B inverse into A inverse. So inverse uh of the matrix AB is always written as B inverse into A inverse. So this is true.

[00:18:09]And if A and B are invertible matrices, they are invertible I mean the any matrix is said to be invertible whenever the inverse exists. So inverse exists is nothing but uh A is a A and B both are invertible they have mentioned means A and B both are having inverses students and hence determinant of A and determinant of B both are never be zero.

[00:18:31]So this is also the true statements. Now we have a question for these two now that is out of this one and two which one is not correct. So for this now we'll take one example. Suppose I have a matrix A now that is A11, A12, A21 A22.

[00:18:50]Then cofactor matrix of this A is always given by that is capital letter A11, capital letter A12, capital letter A21, capital letter A22. Now uh A into adjoint of A we calculate. So this is matrix A A11, A12, A21 and A22.

[00:19:13]And adjoint of A is written students, which is the transpose of the cofactor matrix. So, hence we'll write A11 A12.

[00:19:22]The row we write it as a column. And the second row we'll write it as a now, that is second column. So, now we do that is what? Row to column multiplication. So, here uh this is row, this is row to column multiplication we do. So, first uh first row and first column. So, corresponding elements we multiply. A11 and this capital letter A11 plus A12 capital letter A12. Now, first row with the second column. A11 capital letter A21, A21 capital letter A22.

[00:20:02]Okay. Now, second row first column. A21 capital letter A11 A22 capital letter A12. Now, last second row second column. A21 capital letter A21 plus A22 capital letter A22. Now, next we learned that is we have one theory students in determinants. Whenever you have the elements of the one row and the cofactors are also of the same row. If you observe here, I have a element A11, it's a first row element, and with cofactor that is A capital letter A11.

[00:20:42]So, whatever the elements are there, these are of first row. Even the cofactors are also of the same row. See, A11 A12 first row, A11 A12 first row.

[00:20:53]So, whenever you have the elements and the cofactors are of the same row or of the same column, this result is always determinant of A. And if they are of different, I mean here if you observe the elements are of the first row and the cofactors are of the second row, in that case this result is always zero.

[00:21:11]So, this is again zero, and this is determinant of A. Okay, this is a matrix now. I can take the common common element is here determinant of A. So, here we left with one.

[00:21:24]Uh here we left with zero because uh from zero when we take determinant of A common, then again element remaining is zero. So, this is zero. Here I have taken determinant of A out means this is one, and thus what we got is determinant of A into So, you know this is a square matrix where the diagonal elements are one, and the non-diagonal elements are zero, which is called as identity matrix, and it is denoted by capital letter I. Thus what we got A into adjoint of A, this value is equal to determinant of A into I. I mean that is this one is also correct. So, thus what we can say here is uh the first sorry, the second, third, and the fourth statements, these all are right one, and they have asked you which is not correct, and hence option one is not correct here.

[00:22:13]This is example number six.

[00:22:17]Let us begin with example number seven.

[00:22:20]If A and B are invertible square matrices of order N, then which of the following is not correct? Okay, the first statement they have given determinant of AB.

[00:22:30]Determinant of AB. Okay, determinant of AB students always does. Whenever you have algebra middle that is product, then you can take determinant separately here, that is determinant of A into determinant of B, and thus first statement is right here. Second, suppose they have given determinant of any scalar into uh matrix, then here students the scalar raised to you will get the order of the matrix, and the order they have given N, so you will get K raised to N into determinant of A. And that is the second statement they have given. Hence, the second one is also right here.

[00:23:05]Third one if I talk, that is determinant of A plus B. So, see, determinant can be separately taken, students, only for the algebra, that is multiplication, product. Here they have given addition and they given here that this is same as determinant of A plus determinant of B, which is wrong, okay?

[00:23:24]By taking example also, you can check here. You are not getting, students, determinant of A plus B same as determinant of A plus determinant of B.

[00:23:32]Maybe for uh some matrix, you may get, but not for all. In general, it doesn't holds good. Hence, the third statement is not correct here, and that is option three. And similarly, if I see this determinant of A dash, okay, determinant of A dash they have asked. First of all, remember, determinant of the transpose is always same as the determinant of A, and the determinant of A can be written as always one divided by determinant of A inverse, okay? So, the fourth statement is also right here. And thus, what we can say the not correct statement is that is third one. This is example number seven.

[00:24:12]Beginning with example number eight.

[00:24:14]Here they have given the area of triangle with the vertices 3 8, -4 2, and 5 1. The area of triangle is given, students, which is P by 4. They have asked here what is the value of P. Okay, we learned that is area of triangle formula is 1 by 2 into determinant of 3 8 1 -4 2 1 5 1 1. This value is given already, which is how much? P by 2. Now, let us solve. So, P by 4. Now, let us solve this.

[00:24:46]First of all, you can cancel here 2 1s are and 2 2s are.

[00:24:52]Okay, both are in the denominator. So, now, let us expand this now. Expanding along first row plus minus plus. So, three into bracket uh leaving first row first column first row first column two into one two minus one into one one.

[00:25:09]Next minus eight into bracket leaving first row second column minus four into one is minus four minus five into one five. And next plus one into bracket leaving first row third column minus four into one minus four minus five twos are that is 10. Is equal to how much P by two?

[00:25:29]Now, three this is one minus eight into bracket this is minus nine. Addition of minus four and minus five is minus nine.

[00:25:39]Plus one into bracket this addition is minus 14. Is equal to P by two.

[00:25:45]Now, this is three this is minus minus plus eight nines are 72 and next it is minus 14 is equal to how much P by two. Now, let us simplify this. This is 75 75 minus 14 and that is 61 is equal to P by two. Now, bring this two to the left hand side 61 into two you will get and that is your P value and this is how much 122.

[00:26:17]Thus, P value is that is option three.

[00:26:21]This is example number eight.

[00:26:24]Now, beginning with example number nine.

[00:26:27]Here, they have given that is two linear equations to us. These are called as the equations of straight lines. So, one they have given X plus 2Y is equal to three and then what they have given 2X plus 3Y is equal to three. So, these are the system of linear equations and have asked here these two equations are having with the no solution or unique solution infinite or only two solutions. Okay.

[00:26:54]uh These are actually students the equations of linear equations as I said.

[00:26:58]Linear equations means these represents the straight lines. And we have students straight lines when we talk, no? So, we have a one variety of straight lines which are intersecting. Okay? Second lines are coinciding lines like Coinciding means they are lying on one another. This is a first line and even this is itself is a second line. Or we have a two that is parallel lines. So, these are the three three varieties of lines we have in a plane. So, whenever students the lines are intersecting, no?

[00:27:29]They have unique solution because the both are intersecting at a single point.

[00:27:32]This point lies on the first line also.

[00:27:35]This point lies on the second line also means this point satisfy both the equations. And hence here what we can say is they are having unique solution.

[00:27:44]And whenever the lines are coinciding, so every point is present on the first line also as well as on the second line also. So, we have here infinite points. You can't count. So, what we can say is whenever the lines are coinciding, no? They have infinite solutions. And whenever the lines are parallel, whenever the lines are parallel, no any point is common here.

[00:28:08]And thus what we can say we don't have any solution. And thus we say the lines whenever they are parallel, they have no solution. So, remember intersecting lines have unique solution, coinciding lines are having infinite solution, and the parallel lines are having no solution. Now, we have a question.

[00:28:25]Whatever the lines are given above, whether they are intersecting, coinciding, or parallel lines. Okay. To check these students, remember one thing. Suppose if I'm having a1 + b1 + or is equal to c1 you take. And one more we have that is a2 + b2 is equal to c2.

[00:28:45]Okay? If these are the, uh, lines, suppose I'm having A1X, B1Y, A2X, B2Y is equal to C2. Okay, these lines looks like now, uh, same to same the above lines. Here A1, B1, C1 are the constants. A2, B2, C2 are also the constants. So, here X coefficient is 1, so A1 is 1. Y coefficient is 2, so B1 is 2, like this.

[00:29:09]So, remember, uh, whenever the lines are intersecting lines, no, always A1 / A2 is not equal to B1 / B2. Okay, this happens whenever they are intersecting lines. And whenever they are, uh, coinciding lines, no, what you will get is A1 / A2 is same as B1 / B2 is same as C1 / C2 you will get. Okay? This happens, students, whenever the lines are, that is, coinciding. And whenever the lines are, that is, parallel, no, A1 / A2 is same as B1 / B2, but that is not equal to C1 / C2.

[00:29:56]So, A1 / A2 is not equal to B1 / B2 means they are intersecting lines having unique solution.

[00:30:03]When all three, I mean, this ratio A1 / A2 is equal to B1 / B2 is equal to C1 / C2, then they are coinciding lines. And if suppose, students, these are same, like A1 / A2 is equal to B1 / B2, but that is not equal to if C1 / C2 is there, then these are parallel lines with no solution. So, we check now out of these three, which one holds good here. If I see, students, in this, A1 is 1, B1 is 2. Okay? So, let us calculate first A1 / A2. So, X coefficient we check. Here 1, here 2. So, it is 1 / 2.

[00:30:38]Uh, B1 / B2 we calculate. B1 / B2. B1 is Y coefficient that is two. Here Y coefficient is three. Okay, here only we come to know A1 by A2 value is 1 by 2.

[00:30:49]B1 by B2 value is 2 by 3 and they are not same. So, A1 by A2 is not equal to B1 by B2 means they are intersecting lines are having unique solution and that's what we can say the option two is right. So, this is regarding the example in the ninth. So, with these students that is we are ending the today's video.

[00:31:07]These are the examples they have asked.

[00:31:08]These are the nine examples they have asked in KCET April 2026.

[00:31:15]So, remaining examples we see in the next video. So, we'll meet soon, very soon in the next video students. Till then, take care, stay happy and spread happiness.

[00:31:27]Thank you. Bye-bye.

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