Determinants One Shot | CHSE Class 12 Maths 2027 | Complete Chapter in One Video | By Amit Sir

[00:00:32]Hello. Hello. Hello students. Welcome to classes. All of you determin function rel then we will start today's class all of Okay.

[00:01:09]Okay. Students determinally one of the most easiest chapter.

[00:01:38]hours determinate.

[00:01:59]Okay.

[00:02:01]Determined checklist determine Next minor and co-actor.

[00:02:36]Next minor co-actor. Minor co-actor adjint and inverse. And last topic application equar a1 x + b1 y + c1 = 0.

[00:03:21]I would a 2x + b2 y + c2.

[00:03:40]10th and 11th cross B1 into C2 minus B2 into C1 divided by A1 B2 minus A2 B1. Similarly, minus A1 C2 A1 C2 minus A2 C1 A2 C1 divided by denominator A1 B2 minus A2 B1 got a linear system of linear equation.

[00:04:45]Pair of equal not a value Answer.

[00:05:16]It means a1 b2 a2 b1 is not equal to z.

[00:05:34]A2 B2 A= Zetminant Determinant of a square matrix.

[00:06:05]Square matrix.

[00:06:28]Row number one. Row number two. Row number three.

[00:06:34]column. Column number one, column number two, column number three.

[00:06:42]Matrix say number of rows, number of columns equal col.

[00:07:05]square.

[00:07:21]Determinant of a square matrix is option. Second option.

[00:07:34]Third option determinant of any matrix.

[00:07:47]One by one.

[00:07:55]It means one by one second.

[00:08:23]2x2 matrix 2x2 element data row A2 B2 A1 B2 minus A2 B1 exactly A1 B2 A2 question one by one solve question.

[00:09:30]One by one matrix.

[00:09:37]Answer.

[00:09:42]Next.

[00:10:04]Next question.

[00:10:07]2x2 matrix row square matip it means 2 into 7 14 - 5 into 3 15 it means one similar this one into this one minus this one into this Fore 12 12 - 12.

[00:11:02]Next determinant of 3x3 matrix.

[00:11:16]Let us row number two.

[00:11:28]Row number three.

[00:11:30]Column one. Column two. and column.

[00:11:40]A1 B1 C1 A2 B2 C2 Next A3 B3minally Max number expanding along row one. Expand along row number one. Number one, this is row number one. Row number one.

[00:12:55]A B2 C2 B3 C3 B2 C2 B3 C3 determinant number A1 A1 next B1 B1 A2 C2 A3 C3 A2 A2 C2 A3 C3 determinant 80 plus 80 minus B1 calculation complete. Next. C1 C1 plus 80 plus 80us 80 plus A2 B2 A3 B3 A2 B2 A3 B3 Three determinant Aminant number one expand.

[00:15:18]Row number two. Row number three. Column number one. Column number three.

[00:15:25]Sign.

[00:15:34]Sign convention.

[00:15:38]Sign convention.

[00:15:42]Sign convent.

[00:15:54]80 + 80 - 80 + - - plus + - - plus + - - plus convention plus minus Plus number 80.

[00:16:30]Similarly row number three expand convention plus minus plus Similarly column 1 column 2 column 3 C1 C2 C3 P same condition plus - plus - plus - accordingly column.

[00:17:02]Row number of Determinant of A.

[00:17:38]matrix I'm cross multiply 3 into 1 3 - 5 into 2 - 5 into 2 - 10 - plus 13 matrix Row number one maximum number of row number Determinant 56 23 determinant of 5 6 2 and 3.

[00:18:49]Next complete. Next.

[00:19:06]-163 -1 6 1 3 determinant next complete US number it means determinant of -15 1 Answer number plus this is -4 inside 8 2 into -1 - 2 - 1 2 8 8 - 4 6 7 4 28 28 + 64.

[00:21:01]So final answer 34 question.

[00:21:07]All clear all of you.

[00:21:22]Next question.

[00:21:29]Now minor and co-actor. Next topic basic determinant properties of determinant. It is not in our syllabus.

[00:21:58]M I determinant determinant by deleting. Delete.

[00:22:10]Delete. I row and J column. I row Jetmin.

[00:22:26]And hence we can conclude that is the final answer.

[00:22:30]Let us consider students there is a determinant question matrix form. We have to calculate M13. M13 students the question is written M13.

[00:22:42]This is one this is three. M13 means what? One represent students row. It represent row and three represents column.

[00:22:53]One represents row, three represent column. It means first row and third column. First row and third column ignore. First row row number one for our use.

[00:23:07]This is R1.

[00:23:10]This is R2. This is R3. This is column 1, column 2, column 3. Row number one and column number three. Row number one cortigala. Column number three cortigala. Now we have to calculate determinant of these four answers. That means A2 B2 A3 B3.

[00:23:31]A2 B2 A3 B3 determinant. And how we can calculate determinant? Cross multiply HA a2 B3 minus A3 B2 and then we will get the proper answer. Come to the next question. All of you question try to understand students. The question M11 means what? First row, first column.

[00:23:59]First row cancel out. First column cancel out. What are the remaining elements? B2 C2 B3 C3 B2 C2 B3 C3.

[00:24:13]So it is B2 C2 and B3 C3 determinant. So it is B2 C3 minus B3 C2. This is the final answer. Similarly we can go for M12 M12 calculation be 1 2 means what students? 1 2 means first row and second column. One row row number one two means column number two.

[00:24:47]1 2 1 2 1 means R1 and two means C2 R1 and C2 R1 and C2 cancel out the remaining numbers corner number one A2 C2 A3 C3 that is determinant of A2 C2 A3 C3 A2 C2 A3 C3 whole determinant and uh cross multiply After cross multiplying the answer will be A2 C3 minus A3 C2. I think.

[00:25:27]I think discuss co-actor is denoted by co-actor is denoted by C I Kima A I J C subscript I J Kima A subscript I J I'm co-actor now formula coactor formula just minor migacus to the power i + same answer that will change. Let's see. First of all, the question is A13. A13 last A13 A13 M13 A13 M13 A13 same answer what I mean to say students row number one row number two and R2 R3 this is C1 this is C2 C2 this is C3. First question says one one means R1 3 means column number three row number one and column number three. The remaining elements are A2 B2 A3 B3. So let us write down a2 b2 a3 b3 a calculation a2 b2 a3 b3ic convention what is the sign convention numbers 1 and 3 -1 to the power 1 + 3 -1 to the power 1 + Three means whatus number it means final answer the answer will be a2 b3 minus a3 b2 similarly a calculate a11 this is r1 this is c1 c1 r1 cancel The remaining is B2 C2 B3 C3. So B2 C2 B2 C2 B3 C3 buticient -1 to the power 1 + 1 - 1 ^ 1 + 1 it means -1² and even number then again we can say the number is positive it means -1 to the power square -1 ^ 2 is a positive number + 1 H final answer will be B2 C3 minus B3 C2. Come to this one all of you. Third question. One means row number one and it is two. Two means column number two. Row number one column number two. Row one and column 2 cancel out. The remaining numbers are A2 C3 A3 C2. Answer A2 C A2 C2 and this is A3 C3 AIC -1 to the power 1 + 2 - 1 ^ 1 + 2 1 + 2 is 3 it means -1 -1 meaning it means final minus A2 C3 minus A3 C2 this is my final answer the answer but you can see in the third question minus sign it means there is a sign convention for every question S convention of 2x2 sin convention of 3x3. So it means 2x2 matrix I discuss about 2 plus second minus 80 minus 80 plus this is row number row 2 column 1 column 2 it means a position plus a position co-actor minus similarly co-actor of this position is minus co-actor of this position is plus 3x3 matrix R1 R2 and R3 and on the left hand side it is C1 C2 and C3 so my answers 80 plus - plus plus - - plus - plus - plus determinant for last formula same one one row number one column number one 1 Even direct question plus plusus plus 80 minus Find the minor minor of the diagonal elements. Diagonal elements of the determinant elemental element.

[00:32:49]These are the diagonal elements.

[00:32:53]These are the diagonal elements.

[00:33:05]We have to calculate the minor minor questions.

[00:33:23]And what is the value of I? I value roo<unk> overus1 what is roo<unk> overus1 class 11 complex number complex number 1 i - i then - i 1 i 1 - i and i this is the question The first question number this is the first number. While calculating the co-actor of first number Let us consider row number one. This is R2. This is R3.

[00:34:35]This is C1. This is C2. This is C3. C1.

[00:34:40]C2. C3. First position.

[00:34:43]column. Row number one, column number one. Row number one, column number one.

[00:34:52]Minor it means M11.

[00:34:56]M1.

[00:34:59]First one row meaning row. Second one row meaning first row first column will determinant will be 1 i minus i and i. Yeah. Determinant 80 is coefficient multiply cross multiply it has i - i² we know the value of i² i square -1 -1 and the final answer will be i -1 this is the first question second question diagonal element question Number row second row and column second column. Second row second column it means naming M2 first represent this is for R2 and the second one is for C2. represent R2 and represent C2 R2.

[00:36:26]Okay, fine.

[00:36:37]1 I determinant 1 - I 1 I 1 into I - 1 into I - I ultimately answer will be 1 + I I + I 2 I 2 Iota. Similarly third question row number three column number three that means the number will be m33 M3 it means 33 position number 1 I - I1 right 1 I - I 1 determinant Cross multiply 1 into 1 is 1 - i into - i is - i². The value of i² is -1 - - final answer is 1 + i.

[00:38:06]And Okay. So, next topic coactor using co-actor of the elements of third row.

[00:38:34]Co-actor of elements of third row. Third column. Third column. Evaluate. Evaluate means we have to calculate its determinant value. Determinant value which column third column C3 students this is R1 this is R2 this is R3 this is a C1 this is a C2 this is a C3 so we have to use column number three okay fine column number three sign convention column number three sign convention Column number three sign convention.

[00:39:17]This is R1, R2 and R3. This is C1, C2 and C3. The sign convention for column number three is + - plus - plus - plus - plus the sign convention for third column plus - plus. This is the sign convention we will use. Start first.

[00:39:47]Okay. Fine.

[00:39:50]Expansion plus yand determinant of 1 y 1 z.

[00:40:06]Determinant of 1 y 1 sinus minus z into x while talking about the second number students second number then the remaining numbers are 1 x 1 determinant 1 x 1 z determinant of 1 x 1 Z. Next the third number in column number three it is X Y X Y sign convention plus. So plus XY while talking about XY a number.

[00:40:49]Next we will calculate determinant of 1 X1 Y that is determinant of 1 X1 Y. Now let us try to calculate this one.

[00:41:02]That is determinant of this matrix or we can say that is determinant answer will be y into this is cross multiply z into 1 - y - z into x 8 * z - x next + x into y this is y - x.

[00:41:33]So cross multiply then we can multiply this one and we will get the Next singular matrix matrix meaning a matrix is a singular.

[00:42:13]A matrix is singular if determinant of a is equal to zero. But a matrix singular mat.

[00:42:27]Let us take an example to understand this. Example, let us 3 2 6 matrix determinant of A. Determinant of A. It means question will be 1 3 2 6 whole determinant. The determinant 6 into 1 - 2 into 3 1 into 6 will be 6 - 2 into 3 is 6. 6 - 6 answer 0.

[00:43:04]Yes, my answer is zero.

[00:43:06]Hence, A is a singular matrix.

[00:43:13]Singular matrix.

[00:43:16]Okay.

[00:43:19]What is the meaning of singular matrix?

[00:43:21]All clear. All of you special type determinant students. One is symmetric determinant.

[00:43:36]Another one is skew symmetric determinant. Symmetric determinant symmetric determinant difference.

[00:43:46]Let us try to understand symmetric determinant. Students, let us consider we have a determinant.

[00:44:07]We have to take square.

[00:44:25]Let us consider and here we have another number C.

[00:44:37]symmetric.

[00:44:49]This number is also x.

[00:44:53]Next.

[00:44:57]Let us consider it means next. Let us consider.

[00:45:20]This is called symmetric determinant.

[00:45:23]Can you get the connection? Connection a same number same same and y and a a c.

[00:45:44]These are called as symmetric determinant. Now come to the second part. Skew symmetric determinant.

[00:45:51]Skemmetric determinant. Let us take an example of 3x3. 3x3 row number one, row number two and R3.

[00:46:02]This is C1. This is C2. This is C3.

[00:46:07]The numbers are let us consider first point. Second number is x. This number is minus xus.

[00:46:34]Now if this is y, if this is y then the other one is minus y corner connection. Next, if this one is convey Okay.

[00:47:22]Symmetric determinant symmetric determinant determinant. Circular A B C A second B third Cent determinant.

[00:48:02]Okay. Current determinant important but MCQ those are called asent.

[00:48:41]Okay. Next important topic.

[00:48:58]Adjint of a matrix.

[00:49:03]Important topic. Adjint of a matrix.

[00:49:10]Adjint of matrix.

[00:49:17]First point adjint of a square matrix A is transpose of co-actor of each element of A.

[00:49:36]Aintenote Ajint.

[00:49:55]It means transpose transpose meaning this is the meaning of transer.

[00:50:33]Find the adjint solution 5 -1 1 question.

[00:50:58]This is R1. This is R2. This is C1. This is C2.

[00:51:05]First C1A C1 C1 row of first column. First row of first column actor first column A1 second Number A12 minus position second row first column A2 and one second row second column A2 C11 AACAG.

[00:52:35]Next C12 C12 position. C12 position number plus factor sign convention 8 plus 80 minus. Next let us calculate C21 a position C21 C21.

[00:53:45]Next C22 C22 a position determinant factor number two but but convention convention plus 80 minus 80 minus 80 plus it means 1 - + 2 factor adjint adjint Step number one. Step number oneaculate factor adjint of a is equal to C. C means factor transpose column.

[00:55:32]Row number one. This is row number two.

[00:55:35]This is column number one. This is column number two. Row number one. Row number one. Column number one. Row one.

[00:55:47]column number one. It means 1.

[00:56:00]Simple number - 5A 2 - Second of column number one. Row number two.

[00:56:58]Column number two. Row number three, column number three. This is to calculate adjint.

[00:57:25]question answer question 2 1 8 1 8 Question A1 A12 A21 A22 I sign convent Convention plus - 8 sign convention questionactor C11 C11 C11 Next C12 C12 position. C12 position.

[00:59:10]Answer C.

[00:59:13]Similarly calculate two one position.

[00:59:16]Two one position.

[00:59:27]Next calculate C22 C22 C22 A2 A22 Determinant value everyone.

[00:59:55]Sorry meaning it means finally factor C equal 2 number D minus C minus B AAA adjint of A is equal to row number one. Row number one, column number one and row number two, column number two.

[01:01:01]R1, this is R2. This is C1. This is C2.

[01:01:08]Crow number one.

[01:01:18]Row number one minus b a final answer.

[01:01:50]Question answer question.

[01:02:16]It means adjint of a question attempt.

[01:02:36]Huh? Row column tree.

[01:02:52]Let us find adjint of adjint adjint of number 2 5 - 7 2x2 matrixulate Aal it means 80 First step fiveus and answer completed easy question 2x2 matrix 2x2 matrix Exactly.

[01:04:21]2x2 matrix 3x3 matrix notactor a11 a12 a13 second row 2 1 8 2 8 about 2 3 Third row 3 1 8 3 2 8 H 3 mat sign plus same question.

[01:05:28]It means 3 into 1 3 into 1 3 - 0 into 5 Next C12 C12 into 1 2 5 into 2 minus sign formula plus total answer 12.

[01:06:06]Next C13 C13 C13 a number 2 into 0 - 3 into - 2 - 6 - - + 8 Similar 2us1 Next 22 2 C2 two position C2 two position 8 number 2 + 4 2 + 4 6 Next C 23 C2 A number A number 2 into minus 2.

[01:07:06]Next C31 C31 5 into - 6 - 5 - 6 - 11 answer - 11.

[01:07:42]Next C32 C3 into 2 10 2 into 2 H 4 10 - 4 H 6 Nextulate C3 C33 A 3 into 2 6 6 + 2 Answer import 80 minus and one.

[01:08:48]It means finally factor C 3 - 12 6 1 6 2 - 11 - 6 8 adjint adjint adjint row 1 this is row 2 This is row three.

[01:09:29]Transpose number 126 column number one column number one 3 -26 row number two 162 62 row number three -1 - 68 8 -1 11 - 68 this is C1 column number one column number two column number three row column adjint of A final answer 2x2 matrix shortcut formula 3x3 matrix formula simply 2x2 3x3 factor adjint property 3 answer 3 1 - 11 3 1 - 11 - 12 6 - 6 6 28 perfect answer well and good First property Important property A into adjint of Aalmin into Y3 A1 1 A12 A 21 A22 matrix factor answer C11 C12 C21 C22 adjint Row number one row convert column one column 2 row number column number one.

[01:12:52]It means answer C11 C12. Next C21 C21 C21 C12 C2 A into I. I Identity matrix.

[01:13:26]Identity matrix.

[01:13:30]Identity matrix.

[01:13:43]2x2 matrix 1 0 0 1 2 by 2 3x3 identity matrix 1 0 0 1 0 0 1 element Number one matrix 1 Answer number Question determinant to the power n minus one n is equal to order of matrix.

[01:15:17]Order of matrix number of row.

[01:15:24]Number of row. Number of row.

[01:15:29]Number of column.

[01:15:45]Number of column.

[01:15:51]Same answer.

[01:16:08]Next important point number two most important point number two number five point of view important formula Direct question number MCQ Question number two.

[01:17:16]Step number one, find Column number two C2 C2 expand C2 - plus - plus - plus - plus column number two sign convention - plus it means -1us One number determinant 5 - 2 - 7 4 5 - 2 - 74 answer Answal 6 - 6 of a 5 into Okay.

[01:19:15]value number of row R1 R2 R3 R1 R2 Answer 5 number into a whole determinant power into a it means to the power three into determinant of A.

[01:20:30]5 to the power 3 125 into -6 125 into - 6 125 6 multiply answer 125 6 125 into 6 30 750 750 - 750 answer third question adjint of a whole determinant formula determinant of a power determinant of a power 3 - 6 power 2 - 6 the power 2 answer 36.

[01:21:28]Second question.

[01:21:31]Third question.

[01:21:50]Adjint of determinant 44 adjint of determin entire 36 Next adjint of adjint of a determinant of a to the power n -1 row² determinant of - 6 - 6 power 4 - 6 MCQ questions.

[01:23:16]Next topic. Enter. Inverse of a matrix.

[01:23:20]Inverse of a matrix.

[01:23:32]Inverse of a matrix.

[01:23:39]A power.

[01:23:42]Second point. Matrices whose inverse exist those are called as invertible invertible invertible inverse A inverse A inverse formula Adjint of A divided by determinant of A.

[01:24:18]Adjint of A.

[01:24:31]divide finverse.

[01:25:00]formula adjint of a divided by determinant of a question.

[01:25:09]Okay.

[01:25:18]Invertible matrices whose inverse are exist are called as invertible.

[01:25:25]A inverse adjint of A / A.

[01:25:45]Exist does not exist.

[01:26:01]Number four.

[01:26:03]Number four.

[01:26:07]A is invertible.

[01:26:29]A is invertible when determinant of A is not equal to Z.

[01:26:45]inverse.

[01:26:50]A inverse Imagatip A inverse third into a whole inverse normal number inverse inverse 1 by mat inverse adjint of a divided by determinant of A.

[01:27:56]to prove that prove that matrices inverse inverse into a inverse important propert inverse difference.

[01:28:49]A inverse adjint of a divided by determinant of A.

[01:28:58]inverse of difference by determinant of a Friends, basic bas invertible or not.

[01:29:50]of a not equal to zero.

[01:29:54]Determinant of Aalmin.

[01:30:03]Number one.

[01:30:23]Next final answer of determinant of a notal.

[01:30:44]Yes. 2 is not equal to zero.

[01:30:55]Determinant of a notal it means invertible.

[01:31:01]Hence a is invertible.

[01:31:21]Exactly.

[01:31:26]Next inverse inverse inverse Inverse note inverse inverse adjint of A divided by determinant of A.

[01:31:55]Solution number of A. Step number one. Step number two of Ax23.

[01:32:30]Step number two.

[01:32:33]Step number three. Step number three.

[01:32:40]Answer 1 /min of notal.

[01:33:12]Inverse inverse does not exist.

[01:33:38]solution direct of a determinant of a x - x² determinant cross multiply x - x² - x + x² next adjint of a adjint of a shortcut a position 80 x² x adjint of a direct answer a inverse a inverse determinant of aulate adjint / x + x² x - X Okay.

[01:35:45]Next. Application of determinant.

[01:36:06]Application of determin Area of triangle. Area of triangul.

[01:36:44]A X1 Y1 B X2 Y2 C X3 Y3 area of triangle area of triangle by triangulose.

[01:37:27]Row number one.

[01:37:29]Column number one, X1, X2, X3.

[01:37:33]Row number one. Column number two. Y1, Y2, Y3. Row number two. Y1, Y2, Y3.

[01:37:43]Number one.

[01:37:46]Number three.

[01:38:04]Area of triangula of triangle is always positive.

[01:38:31]Area of triangle is always positive.

[01:38:39]First point, second point. If area of triangle is given, area of triangle is given in the question.

[01:38:53]given in the question area of triangle question given take it as take it as plus area of triangle area of triangle questionf Answer question.

[01:39:41]Find area of triangle whose vertices are 2 3 5 4 6 Area triangle A B Cra value.

[01:40:24]Five.

[01:40:33]Y side. Fourth. Five side. Four. Seven.

[01:40:50]Side. Six.

[01:40:52]Seven last number 2 - 6 + - 3ip - 7 + 30 - 74 28 solve solve 1x 2 - 2 - 2 - 4 8 - + 6 8 + 2 total Answer 6 + 2 8 8 - 4 4 / 2.

[01:42:01]Next square unit square units.

[01:42:38]Sumitra question.

[01:42:58]Next question. Nextity.

[01:43:01]Next application. Point number two.

[01:43:16]straight line.

[01:43:23]A comma ba C are collinear.

[01:43:44]If area of triangle ABC= ABC1 Y1 B X2 Y2 C X3 Y3 X3 Y3 last one 1 total determinant value 0.

[01:44:34]It means X1 Y1 1 X2 Y2 1 X3 Y3 1 X1, X2, X3 Y1, Y2, Y3 1 Amin, B and C are straight line.

[01:45:34]Check point A Cordin A B C Point B coordinate B comma C + A coordinate C A + B area of triangle A B C 1 area.

[01:46:40]Next B C A Brow Number Nip.

[01:47:10]C + A - A + B A first part next C B minus C.

[01:47:38]Next one discuss a cross direct answer a b + b² - c² - a c a c minus - b 8 b + c into b minus c b² - c² a minus a c + b² - c² multiply a c - a - b² + c² + a b - a c + b square - c square by cancel out final answer.

[01:48:54]Hence a b c are collinear.

[01:48:58]A comma b comma c are collinear.

[01:49:08]All the three points are collinear.

[01:49:16]Exactly.

[01:49:23]Next.

[01:49:29]Next.

[01:49:30]Equation of straight line. Equation of straight line. Application. Application number three. Equation of straight line.

[01:49:54]Equation of straight line.

[01:49:57]Equation of line pass through the points.

[01:50:21]Let us consider a Ainordinate X1 Y1 X2 Y2.

[01:50:39]Imagine let us conser Y1 X1 Y1 1 X2 Y2 1 Answer 0 Answer Z of straight line of straight Find equation of straight line pass through data point 2us 3 1 A value 2us 3 let us consider a point Y means X Y 1 8 2 80 - 3 80 S 1 81 1 8 Z equal Row number one number one. Next 2 1 2 into 1 2 1 into 1 2 + 3 2 + Right hand side= 0 - 4x= 0 = 0 equation of straight Straight line equation. A straight line.

[01:54:37]Except question 100% topic coverage question.

[01:55:03]Next, system of linear equation.

[01:55:30]Last topic.

[01:55:37]Last topic. Important. The most important topic in this chapter. A topic long question of equation Minute.

[01:56:18]Exactly.

[01:56:25]Last topic.

[01:56:27]System of linear equation.

[01:56:32]2 minutes.

[01:56:36]Come on. Salah.

[02:01:16]Hello. Hello. Hello. Students, start.

[02:01:19]Last topic. One of the most important topic determinant topic. System of equation.

[02:01:28]System of equation.

[02:01:41]A1 + B1 Y + C1= D1 value.

[02:01:56]Find the value of Find the value of X Y.

[02:02:18]point.

[02:02:36]Determinant vales to calculate for A1, B1, C1, A2, B2, C2, A3, C B3 C3 equation X Y D1 D2 D3 capital A into X= A x= a x = b = a inverse into b x= a inverse into a inverse a inverse = A inverse into B inverse adjint of A divided by determinant of A into B.

[02:05:17]X= B into A inverse.

[02:05:36]A inverse inverse determinant of a notaletal determinant of matrix determinant value.

[02:06:14]A is a singular matrix.

[02:06:32]adjint of A.

[02:06:44]Answer value.

[02:06:59]Yistent solution.

[02:07:11]Consistent solution.

[02:07:17]Unique solution.

[02:07:20]Unique solution.

[02:07:31]Adjint of a balment adjint of a into balent of a into b is not equal to zero Infinite number of solution.

[02:08:06]Infinite number of solution.

[02:08:25]Determinant of a bal inconsistent inconsistent no solution.

[02:08:39]Inconsistent solution xriv solution.

[02:08:54]Trivial solution.

[02:09:07]Consistence of a=ist.

[02:09:40]No solution.

[02:09:53]The solution is consistent.

[02:10:00]It is a trivial solution. Trivial solution of a notal.

[02:10:07]Inconsistent determinant of a= A x = b1 b1 c1 A2 B2 C2 A3 A3 B3 C3 rle dx dy dz x y d1 d23 clearly Determinant of Aat a mat of A.

[02:11:35]Next.

[02:11:37]DX DX DXIC A1 A2 A3 D1 D2 D3 It means B1 C1 B2 C2 B3 C3 determinant value matrix determinant next dy A1 A2 A3 separator C1 C2 C3 D1 D2 and D3 dx a1 B1 1 A2 B2 A3 B3 D1 D2 and D3 and Xulate dx divided by d I dy / the following system of linear equation.

[02:14:10]Solution startic 1 1 2 -1 3 2 -1 x YZ equal to 41 matrix A matrix X matle Dalmin 1 1 1 2 -1 3 Next 3 2 -1 determinant 1 1 - 6 next -1 8 - 2 - 9 next + 1 4 + 3 simpl Determinant plus 11us total 13.

[02:16:20]Calcul - 8 - 2 - 9 + 3 + 11 dul dx dx next page dx Next 1 2 3 1 2 3 4 1 41 dx dx determinant 41 question 1 -12 1 -12 2 ATS 1 3 - 1 1 3 - 1 already five marks question number four 1 - 6.

[02:17:50]Next -1 -1 - 3.

[02:17:56]Next + 1 2 + 1 simplify - 20 + 4 + 2 + 3 -3 similar Similarly, dy dx dy dy y 2 1 2 3 4 1 simplify solve. Try solve number one expand next 2 - 3 solve final Answer 39.

[02:19:27]Similarly, DZ DZ position 411.

[02:19:42]DX position 1 2 3.

[02:19:45]DY position 26 answer. Final answer for dx / dxus value 13 -3 / 13us yulate dy / dz divided by dy / y 39 / 13.

[02:20:38]Similarly, DZ / D 26 / X=1 Y = 3 Z = 2 by Answer.

[02:21:23]Next. Matrix method. Matrix method.

[02:21:27]Discuss adjint of A divided by determinant of A.

[02:21:59]left side.

[02:22:03]It means right answer.

[02:22:19]A bimp determinant of A value B / determinant of A Z C / Amin.

[02:23:15]Okay, let's try to solve one question all of you.

[02:23:22]For what value of lambda question?

[02:23:28]It is not the same question.

[02:23:34]For what value of lambda the system of equation system of equation given a this system of equation do not possess a solution. It means lambda value put a system of linear equation answer we will not get any solution. No solution we will do not possess any solution. What is the meaning of do not possess any solution? Means it means students it has got no solution and no solution condition no solution condition let us check no condition solution plus students here we have discussed students do you remember no solution come to this page see all of you no solution no solution that means deter Determinant of a=0. According to the concept given in our book inconsistent or no solution determinant of aal consistent trivial solution that means determinant of not equal to a concept that means determinant of a is not equal to zero. Simple question.

[02:25:00]First of all, we'll try to simplify this question all of you. So I can write down the question as solution come to the solution part 1 1 this is 4 lambda and minus lambda and this is 3 2 and minus 4.

[02:25:22]This is matrix A.

[02:25:25]Then this is X Y Z X Y Z is matrix X and equal to number 6 0 and 5.

[02:25:39]This is matrix B. A X = B and it is given in the question question record given students. It has no solution. No solution implies implies determinant of a is equal to zero. Now let us try to solve this one everyone. Determinant of a equal to zero. Determinant of a calculate this is 1 1 1 4 lambda minus lambda 3 2 and - 4 this determinant is equal to zero. Let us try to expand this along with row number one. row number one expand and let's see the solution.

[02:26:24]So we can write here students 1 into -4 lambda + 2 lambda - 4 lambda + 2 lambda -1 this is - 16 + 3 lambda then + 1 this is 8 - 3 lambda and this is equal to solve and then check what is the value of lambda lambda value okay let us simplify this one - 2 lambda this is + 16 - 3 lambda this is + 8 - 3 lambda =0 so lambda answer students this This is first part. This is the second part.

[02:27:26]This is the third part. So - 8 lambda and this is 16 + 8.

[02:27:37]16 + 8 is 24 is equal to zero. Hence we can simplify this one. Simplify lambda answer three.

[02:27:47]So finally we can conclude that the lambda value is three. Lambda value three. put lambda value zero that means this is no solution condition satisfy I think all of you clear this one no solutioncept now let's go to the next one all of you examine the consistency and solve again this is a five mark question all of you exam fix determinant true question long question.

[02:28:26]Let us try to solve question.

[02:28:30]First one is consistency. Second one is solution. First of all, we have to check the consistency. How to check consistency students? First of all, let us go to the solution. I'm question 1 -1 1. This is 2 1 - 3 1 1.

[02:28:50]This is matrix A. Then X Y Z is equal to 4 02.

[02:29:02]This is matrix A. This is matrix X. This is matrix B. So we can say A X = B. Or we can write X = A inverse B. A position interchange. as I told you a inverse b into a inverse.

[02:29:22]Okay.

[02:29:31]First of all we will calculate determinant of a. All of you determinant of a which is 1 -1 1 2 1 - 3 1 1 and 1. Let us calculate the determinant.

[02:29:54]Let us start calculating the determinant everyone.

[02:30:02]Determinant not equal to zero. 1.

[02:30:08]This is 1 + 3.

[02:30:12]Then + 1. It is 2 + 3.

[02:30:18]Then + 1. This is 2 - 1. So after solving the answer is 4 + 5 + 1. Answer 10. Can I say determinant of a is not equal to zero? If determinant of a is not equal to zero, we can say it is consistent.

[02:30:39]It is consistent. It means it exist.

[02:30:44]That means inverse exist. Consistent meaning inverse exist.

[02:30:52]We don't know.

[02:31:02]Okay all of you we can solve this question inverse number one co-actor co-actor step number two I'm adjint of a adjint of a means Step number three of multip then finally number four that is x = a inverse into b a inverse into b that means x y right side one adjint of A into B divided by determinant of A.

[02:32:05]Okay.

[02:32:12]Then adjint of AI determinant of Aswus 10 and 10 20 - 10 and 10 answer and solve after solving answer 2 this is -1 this is one 8 left side x y then we can conclude this is the method but you know the method Clear all of you. Long A + B + CA is equal to Z.

[02:33:46]question simplify a = x.

[02:34:01]Second equation simplify.

[02:34:06]If we simplify the second equation, the answer will be bz minus bx = y. And the third equation simplify.

[02:34:25]Third equation simplify answer.

[02:34:29]CX minus C Y x - a y + az = 0.

[02:34:53]Okay.

[02:35:03]BZ =0 and last equation simplify the answer will be CX - C Y = 0 right hand side.

[02:35:25]So can we say it is another type of equation students in system of linear equation right hand side number then this is called as homogeneous equation homogeneous these are called as homogeneous equation homogeneous Okay. Fine.

[02:35:52]This is the part we have to prove.

[02:36:01]Step number one.

[02:36:16]AL students - A a - B -1 B C - C - one this is matrix A determinant 1 A1 1 + b c next + a into b - b c b minus b c next - a into b c plus c and determinant of Determinant of a= for homogeneous equation determinant of a is equal to zero.

[02:37:21]A questionnet x Y X = Y =0 and homogeneous equal to solve 1 + a + b c + ca is equal to zero. This is the final answer and hence proved.

[02:38:34]Hence it is proved.

[02:38:37]So I think Major difficulty questional.

[02:39:07]Okay students I think class 12 all the chapters we have to complete we will complete data chapter complete relation and function determinant in the next class students we will discuss about matrices stay with us connected before the exam syllabus complete MCQ question discuss long question discuss long way to go. We need your support that will support us that will help us and so that's all for today all of you. See you in the next video all of you. Thank you everyone.