Tiger heat. [Music] [Applause] [Music] [Applause] [Music] Hi deers, welcome welcome back to it's me Shahidan plus two mathematics chapters. for. Yes. Yes. Yes. Yes. Yes. 50% questions. So matricesandom columns and columns. What is the order of this matrix? Yes. Rectangular order. Yes. Yes. What is the order of this matrix that is 3x2? So M by N is the general form of a matrix. matrix for any number of elements. Okay. So 100% question. Same question. Okay. Okay. Sad smile. Yes. Okay. A I J for A I J for example first row first column. So this is A11 first row first column. Second row. So this is A2. Second row. Sorry. So very important 2x2 matrix 2x2 matrix for 2x2 matrix column 2x2 matrix for A11 A12 A21 A22 A11 A11 A11 I sorry 1 + 2 into 1 2 into 1 2 / 2 3² 9 by 2 that is equal to 4.5 Okay. A IQ1 plus J 2 2 into 2 4 square by 2. So 1 + 1 5 - square 25 by 12.5 set Max okay so 12.51 what is a 21 that is 21 I2 J1 2 into 1 that is 2 2 + 2 square by 2 that is equal to 4 square 8 and a22 that is 2 + 2 into 2 4 by 2 6. So s into question. Okay. So answer. So final answer is third question answer = first answer 1 + 2 3² 4.5 A12 + 12 12.5 A 21 A22 minimum max okay fine Yes. Yes. Yes. Yes. Okay. Next. Next. Next. Equality of matrices. Equality of matrices. Mat condition number one. First element. First element equal second element equal third corresponding element should be equal decel corresponding element should be equal 2x2 2x2 by 2 by2. So, y = 4, z = 3, x = 1. Okay. for answer. Okay. X Y X Y X You multip value two possible. Okay. Okay. So 5 + = 5 + = 5. So z= 5 - 5 that is equal to 0. If it is perfect say perfect. More examin. for Max. Max Definitely Finance management. foreign. Okay. X + 7= X + 7 = 9 2 + Z = 5 2 + Z = 5. So z = 2 - 5 that is equal to 3. And last y + 3 = 7. So y = 7= x value four and sorry y value four value three. Okay. So previous. Yes. X + Y = N and R= N 1 + Y = 6. So y = 6 - 1. Okay. x + y= x + y = So 5 - 1 that is equal to 4. Okay. Yes. scalar multiplication. [Music] number important. So first yus okay Plus minus on equal to Yes. So, x + 3 = 7. x + 3 = 7. So x = 7 - 3 that is equal to 4. So x = 4 by 2 that is equal to 2. element. Okay. A22 A22 A22 A22 B2 C22. So y - 4 = y = So y= perfect. Yes. Yes. Yes. Yes. Yes. Yes. Perfect. Okay. Fore question. Yes. Equ x + y + x - y + y - y that is 2 x + yat Right side. Right side 2x that is equal to 2x gaming. Okay. So x 2 1x2 into 1x2 into 0 into 1x2 and 1x2 into x mat 4 4 04 - 4 44us. Okay. X + Y X + Y - X - - Y. So plus 2 yus moon rus puja pujus minus on the path. This is equal to 2 y. This is equal to 2 yus. Okay. So, so y= y = 2x 2 = 1 - 4x2= - 2 0 by 0 10 by 2 that is equal to 5. Yes. N - 1 + 1. Yes. Okay. Okay. square x sin square x + cos square x cos square x sin square x + cos square x sin square x + cos square x cos square x + sin square x cos square x plus sign. Yes. Yes. Yes. Yes. Yes. Okay, I hope it is clear. Identity matrix matrix multip. So find matrix matrix 2x 2x = 1 0 - 3 2us plus yus y 3 2 1 4 So 1 - 3 that is equal to -2 0 - 2 that is equal to -2 - 3 - 1 that is -4 and 2 - 4 that is equal to -2 2x what is the value of x -1 -1 -1 - 2 and - 2 - 1. So this is x. Okay. Okay. Okay. Okay. Okay. If it is okay, say okay. If it is okay, say okay. fort. forch. foreign. foreign. Okay. for curiosity. infection. Minimum for option. Okay. Okay. Multiplication of a matrix very very important. Yes. Find Americ 2 by3 Matx multiplication. matrix multiplication possible. For examp 2 by3 Number of columns in first matrix is equal to number of rows in second matrix. possible. What is the order of what is order of AB? That is 2x3. Seven CR 7. Okay. First column. First row. First column. First row. Second column. First row. Third column. A1 12 sorry 9 into 7 A1 A1 first row first A1 first row second column A2 into Okay. A11 A12 A13 A11 A12 A13 Okay. second row. Okay. for final answer. Okay. Yes. Yes. Yes. Yes. Okay. Okay. So I hope it is clear. So Okay. Okay. Okay. Transpose of a matrix. Transpose of a matrix. Matrix. Transpose 3x2 matrix 2x or a to a transpose 3 roo3 1 0 - 1 by transpose. Okay. Okay. A whole transpose equal to A transpose B plus B transpose. A whole transpose A transpose equal to A transpose into Bose. A B transpose always B transpose into Apose and transpose. Okay. So important. So Okay. Okay. question. Okay. So, construct a 2x2 matrix. 2x2 matrix 2 I + 12. Yes. Yes. Yes. Yes. Yes. Yes. important and a transpose different level. Okay. 2x2 matrix. So 2x2 matrix A1 matrix A1 1 A1 2 A 2 1 A 22 2 I I 2 I + 2 into 1 + 1 that is 3 2 + 1 that is 3 2 into 1 2 + 2 that is 4 2 into 2 4 + 1 that is 5 2 into 2 4 + 2 6 Okay 5 A transpose A transpose 3 4 5 6 3 4 5 6 transpose 3 4 5 6 A plus Apose Apose A minus transpose. Okay. and symmetric. And symmetric minimum 2 - 3 - 3 7 in the transpose. Okay. What is the transpose of this matrix? 2 - 3 2 - 3mm. Okay. Okay. Okay. 0 - 7 7 15. What is the sorry 15 zero? Happy. Okay. So 0 - 7 0 0 - 7 A matpose a transpose symmet element always symmetrical element diagonal element. What is the answer of this question? symmetric matrix if it is a symmetric matm so x - 2= 0 so x - vector till and if the matrix is skew symmetric symmetric condition a transpose 0 3 a b 0 - 2 3 2 0 - 0 0 - b - b - 3 - 3 negative 0 - 2 - a 2 0us a transpose a= apose US equal. So minus balus 3 a= a a= a= a= z a + a=0. So a= a= a= minus a a= minus a. Sous a + a=0. So 2 a=0 a= 0 3 a 0 0 3 a b 0 0 - 2 b 0 - 2 3 2 0 3 3 2 0 So negative - 0 - b - 3 - 3 0 - 2 - Okay Okay. Okay. - a= 3. So a= 3 - a= 3. So a= 3. Okay. Aposmus Apmingus 3us Okay. So important symmetric and symmetric Yes. symmetric and symmetric sum of symmetric and symmetric very important and Q matrix A 1 2 3 4 1 2 3 4 Yes or symmetric. Yes or no? Is this symmetric or not? Yes or no? Ape 1= 1= 3 1 + 1 2 2 + 3 5 3 3 + 2 5 4 + 4 2 58mm symmetric symmetric bosm A + A transpose A + A transpose transpose symmet A transpose symmet [Applause] 1 - 1 0 2 - 3 - 1 3 - 2 1 and 4 - 4 0 mat transpose 0 - 1 1 0 symmetric very important Apose is symmetric and A minus A transpose is Q symmetric. A plus A transpose is symmetric. A minus Apose Q symmetric. Very important. Very important. Very important. Q symmetric symmetric and symmetric symmetric symmetric symmetric symmetric sum of symmetric and symmet Okay. Okay. A transpose. So equation very important half into a plus a transpose plus a minus a transpose symmetric symmet So P + Q= A P into A plus A transpose Q into Apose. Okay. So express a as the sum of symmetric and symmetric symmetric symmetric. So half into a + a transpose plus/ into a minus a transpose equal to express half into 6 912 + half into a - 8 0 - 1 1 0/ into 6 3/ into 9 4.5 Half into 9 4.5 Half into 12 6 plus half into 0 0 half into - 1 - 0.5 Half into 1 0.5 Half into 0 0 That is prom a transpose plus half into a minus a transpose. Next 2x2 matrix. Same question. 3x3 matrix. Okay. Okay. equation half into a plus a transpose plus into a minus a transpose equal to a transpose 1 4 - 1 2 5 4 -1 - 6 Apose Ape Ape into a transpose What do you mean? -pose. Okay. Yes. If it is okay say okay a transpose into a transpose subtract. minus render minus minus minus USUS symmetric sorry. So sum is equal to/ into a + a transpose plus/ into a minus a transpose equal to - plus - 2 into a transpose US. Okay. 14US one 14us 4 6 - 2 2 So okay 254 254 254 - 1 - 6 3 - 1 - 6 3 Yes perfect. So that is equal to full mark. Full mark. Okay. Okay. Import Angus let a= a iender. 2x3 matrix. 2x3 matrix 2x3 row A11 A12 A13 A 2 1 A22 A23 Okay. So 1 + 1 2 1 + 2 3 1 + 3 4 2 2 + 1 3 2 + 2 4 2 + 3 5 A into Apose April. and A + B always. Sorry. A transpose plus B transpose. A plus A transpose. Apose always. So Apose 2 3 4 3 4 5. So multiply a into a transpose A into A transpose equal to N 2 3 4 3 4 5 3 4 5 into 2 3 4 3 4 5 Let A into Apose equal to C2 possible 2x2 matrix. So 2x2 First, second column. for A into Apose. A into A transpose transpose equal A into C= to C transpose or A into A transpose A into A transpose transpose symmetric Perfect. Yes. For any square matrix. Prove that. A transpose Apose Ape Let Apose Bose Apose taking transpose on both sides. B transpose equal to A + A transpose whole transpose. Taking transpose on both sides. Transpose transport on both matrix both sides both sides. transpose transpose A transpose plus B transpose. So you know a plus a transpose transpose A transpose plus A transpose whole transpose B transpose equal to A transpose plus A transpose transpose A transpose A transpose Apose A balpose Apose. So this is equal to Boseal B. So this is symmetric. Then B equal to A minus A transpose. C= A minus A transpose AU transpose transpose. So C transpose equal to A transpose minus A transpose transpose. So C transpose equal to A transpose minus Apus that is equal to 3 - 4 - no need of examples. You have to prove in algebraic. Okay. A into a A cos theta sin theta minus sin theta cos theta cos theta sin theta minus sin theta cos theta cos theta into cos theta cos² theta cos square theta cos theta cos square theta sin theta into the sin square theta cos square theta - sin square theta what is cos square x - sin square x that is cos 2x that is cos 2 theta cos theta into sin theta sorry cos theta into sin theta cos theta sin theta cos theta sin theta plus sin theta into cos theta sin theta cos theta sin theta cos theta sin theta cos theta minus sin theta into cos theta sin theta cos theta cos theta minus sin theta - theta cos theta sin theta into sin theta sin square theta and cos theta cos² x - sin square x 1 cos 2 x 2 cos theta sin theta Sin 2 theta sin 2 theta equations equations xt cos theta sin square theta that is equal to cos 2 u theta cos theta sin theta cos cos theta sin theta that is 2 sin theta cos theta that is sin 2 theta cos theta sin theta cos theta sin theta that is 2 sin theta cos theta sin theta sin theta 2 sin theta cos so minus sin 2 theta sin 2 theta and cos² theta minus sin square theta that is cos 2 theta cos 2 theta sin 2 theta cos theta sin theta sin theta cos theta okay A square - a + 1 3 - 2 4 into 1 3 - 2 4 1 into 1 1 + 3 into -2 - 6 1 - 6 - 5 1 into 3 3 + 15 - 2 into 1 - 2 - 2 - 8 - 10 - 2 into 3 - 6 - 6 + 16 that is equal to 10. Okay. So a square - 5 5 into 1 5 5 5 into 3 15 5 into - 2 - 10 5 into 4 20 and 10 I yes 1 0 1 10 0 0 10. So - 5 15 - 10 10 a² - 5 a 5 5 15 - 10 20 + 10 I - 5 - 5 that is - 10 - 10 + 10 0us 0 0 - 0 0 - 10 + 10 0 0 + 0 0 10 - 20 - 10 - 10 + 10 0 Good answer if it is okay say okay sir a inverse Determine definitely answer inverse inverse into a A inverse into a square and a n a inverse into a square and a inverse into a inverse into a square - 5 into a into a inverse + 10 into i into a inverse equal to 0 into a very important inverse into a inverse always. Okay. So a minus a into i - 5 i plus 10 into i inverse inverus I. So what is I in? So 5 I - A equal to 10 A inverse. So 5 0 5 0 5US A inverse what is a 1 3 - 2 4. So subtract 5 - 1 4 0 - 3 - 3 0 - - 2 2 5 - 4 1 A inverse 10 A inverse inverse 1 by 10 4x 10 - 3x 10 2x 10 and 1x 10 2x 5 - 3x 10 2x 10 1x 5 Final answer. for a square - 8 ss. Okay. So deers now we are going to conclude lesson are you there any addition question. Okay. So now we are going to conclude our lesson. Wish you all the best guys. It's me. Thank you miss. Okay. Okay. Okay. Okay. Okay. Byebye. Determine because Yes. Fully satisfied. throughout [Music] ready. So sit So yes ready. Okay. Ready start. for so ready. Okay. Nidish creations 96% E1 by 1 matrix, 2x2 matrix, 3x3 matrix, square matrix number, particular value. Fore number of rows. Number of columns mat 1 by 1 matrix. 2x2 matrix, 3x3 matrix, 4x4 matrix, 5x 5 Foreet square. square 4x4 by 4x4 5 max 3x3 4x4. Max 2x2, 1 by 1, 3x 3. Okay. Determinant. Determinant. To every square matrix to every matrix to every square is of order n One by one matrix, 2x2 matrix 3x3 by one. One by one. Okay. order. We can associate a number complex number. determinant of the square matrix. Okay. 1 2 3 column. So we know that it is a 2x2 matrix. 2 by 2 by 2. Very good. Very good. for each. It is determinant of a determinant of a Determined square bracket Square bracket bracket. D of A D of A determinant of the matrix A. Determinant of 1 2 3 4 because our matrix is 1 2 3 4 and E of A. Determinant of the matrix. foreign. Okay. foreign. So first Correct. It's very simple. Option A. Option B. Determined is a number associated to a matrix. Option B. Option C. Determined is a number associated to a square matrix. Square matrix. associ. Third. Okay. Determined of a matrix of order one. So one for example it's very very simple number of elements 1 by So finally determinate. So we need determinant of seven. Continue. Next example. B is equal to 1 by determined of the matrix B. It is important. So the answer is okay of one by one matrix. No more examples. Two by two. Two by two. Two by two. Two by two. Matx A is equal to A= 8 4 2 6. Okay. 8 4 22x 2 by 2 by 2. So I need determin of so for means multiply Okay. 8 into 2 16 and 2 into 4 16 - 8 means multip First question evaluate determinant of 2 4 - 5US one row columns. Of course it is a 2x2 I need determinant of 2 4 - 5 - 1 is 18 18 2us it is -2 into okay but 2 into -1 - 2 minus - so - 20 + the answer is 20 - 2 = Otherwise you will get full marks. for max 2us US overident. Fore by two determin. question. My new evaluate. Next number Fore cos theta into cos theta. Multiplication sin theta into sin theta square. It is cos square theta cos square theta minus sin theta into minus sin theta sin theta into sin theta sin square us. So cos square= very important question. So function number evaluate the value of determinant of x x -1 x + 1 X into X -1 into X + 1 or we can write X + XUS X² square us. Okay. A into a square - b square a + b into a² - b square. So x + 1 x + 1 into x - 1 a² - b square x²us 1 x² Okay x² So minus of x²us of - 1. So minus of - x²us here. The correct answer is one. science. complete. Okay. So we are moving to if a= Okay. If a= Sorry missal mat 42 then show that determinant of 2 a= 4 into determinant of a okay of volume four into volume determined of two. Okay. Determin A given it is 1 2 4 [Music] 2 into 4 2 4 8 2 which is determin. Okayus 32 It is -4. The answer is -4. 8 - 32 is -4. Very good. Okay. Four into determin of a determin of a determin of so 1 into 2 4 into 8 2 of a so 4 into a -4 into determin of a random exactly same. So here we can see that determinant of 2 a is equal to 4 into determin of a. So we can write hence proved final step. Ready up. Ready. One more. So two question find the values of x if determinant of 2 a determinant of 2 4 51 is equal to determin of 2x 4 6xing elements for finding the value of x corresponding elements because so 2x2x2 Ready usal to 2x 4 6 but 2x into x. It is 2x² x² into x 2 x²us 24. Yes. Only one value. Okay. 2us 2x² - 24 - 18 + 24 = 2x². It is r= 2x² 6 = 2x² 6 by 2 = x² 6 by 2 3 = x² or we can write x²= 3= x= Yes very plus or minus root. It is plus or minus roo 3 possible x = roo3 x=us roo3 these are the two values of x values find the values of x find the value of x Common sense next [Music] column 3x3 matrix 3x3 Max starting. Okay. So, so alternate minus Plus plus - plus - plus - plus. Okay. Plus, plus important A11, A12, A13. First row, A21, A22, A23. Second row, A31, A32, A33. Third row. 3 by3 just by determined find determined finding for finding the value of determinant or you can use second row or third row first column. Second. Third. important factor. First row firstus plus Sure. 2x2 find 3x3 because Fore which means 3 into 0 1 into 0us. Okay. First it is wrong completely - 1 - 4 into 0 - simple It is it is + 4 + 4 into -1 into 4 into 12. So -4 -2 -4 -1 into 1 - 1 4 into 3 12 subraction. So - 4 into -1 -2 parame is -3 4 into -3 is equal to First We should expand clear. Third columnus. Okay. Third column. Third -1 into 1 -1 4 into 3 12 -1 - So - 1 into 1 1 4 into 2 8. So 1 - 81 last 1 - 1 - 2 3 - 2 0 for completely actually because third column is expanding. Fore into 4 into -1 - 12 it is equal to 4 into -3= 52 Okay. So, one more determined find. foreign. Yes. Relax me. foreign. So 3x3 first. So 1 into 1us - 2us4us actually. Okay. 1 into 1us - 2 into - last it is + 5. Very good. 1 into 3 - 2 into 1 it is 3 - 2. Okay. 3 into 1 - 1 + 1 7 + 4 into 1 - - 4 - 1 1 + 4 5 4 into 5 5 1 5 5 into 3 - 2 1 3 into 7 21 4 into 5 20 5 into 1 46 is the right answer. Very good Ashik. Yes. Very good. Very good. Very good. Super. 46 is the right answer. 42. Okay. Ready? You need more questions or not? for. So next properties first determined of Kalet actually important order of matrix. Our determin of K is equal to K into a propert. Okay. Next. If determined of a five by3 of value K to Net of Achment of A 2 into 5 2 8 into 5= Okay. A matrix of order two. And it is given that determin of K is equal to 64 into determin. Which of the following is the value of K of K A= K into Kore? Definitely 64 to n4 to So k is equal to plus or minus roo of 64 plus 8= 8= Option option C right answer. Yes. Very good. Okay. Here right answer. So 490 mark. So a= a = 13 41 matrix. Then find determinant of 3 a transpose. for determined of A determined of a transpose same value Plus two plus serious for a for [Music] Majority. So we have time. foreign. So a=3 determinant of 3 a transpose determinant of 3 a transpose determinant of k a= k to n into determin of 2x2. Okay. Okay. to determined of a transpose. determined of a. So for finding determin of a transpose determin so 1 into 1 1 - 4 into 3 12 - we know that determin of a determin of a transpose 9 into 11 the Answer is - 99. Okay. So moving to next topic. Next topic area of a triangle for finding the area of a triangle. X1 Y1 X2 Y2 X3 Y3 equation Using determinance of [Music] Why? But why? Okay. for delta area in is negative. Whatever the answer is in the case of area finding of triangle. Okay. P. The vertices of a triangle are 0 2 03 and 4 6. Then the area of a triangle is dash equation half into determinant of 023 4. Area is equal to half into minus plus minus plus minus plus minus plus the minus. Four plus four into 1 - 3 into 1 2us 3 Okay into 4 into 2 - 3 2 - 3 but - 4 into - 4 into -4. So half into 2. So the answer is so ofus. So the answer is option B magic. Okay. Answer is option B right answer. YouTube. Okay. So, next question. Find the equation of the line joining A13 and B 0 using determinance. Okay. Equation of the line. Area of a triangul. Okay. point. X Y points points. If these three points are lies on the same line, we can call it as collinear points. points. Okay. Formation of triangle. Area of triangle. Equation of line. area of triang because we know that these three points are we know that area of triangle okay XY Very good. Very good. Very good. It's very simple, guys. Half into second. Second row sign plus - plus - plus - plus - plus second row minus plus secondus - 1 y - 3 into x 1 y 3 into x 3xal half by Half -1 into y - 3x -1 - 1 into y - y - 1 into - 3x y 3x i= - y + 3x or we can write 3xus y= 3x - y = Yes questions. happy. So sorry. Okay. Find the value of K if the area of the triangle is four square unit and the vertices are -2 0 4 and K. Yes. Find the values of question. Max find the value of area of triangle 4 into determinant of - 2. Okay. value. Areus for finding the value of area plus or minus half. So finally plus so - 2 into 4 into 1 4 - k into 1 Okay. plus - 2 into 4 - 4 into it is plus or minus Yeah. - 2 into 4 - 8 - 2 into - k + 2 is equal to plus or - 8 1. So - 8 + 2k plus or - 8 plus orus 8 - 8 + 2kus 2kal 16 2 k= 16 by 2. Okay. 2k = 8 + 8= very good. You are right. Find area for finding the inner value plus foreign. Okay. Next important. Okay. for it's very simple topic. So miners import. Of course, very important. Okay. Okay. Okay. Ready. So question find the minor of the element six in the determinant minor of the element. Six hours 3x3 A1 1 A12 A13 A21 A22 A23 A3 1 A32 A33x3 for So minus capital M11 M12 M13 First row minor M21 M22 M23 second row minor M31 M32 M33 row minus C or A A11 A12 A13 A21 A22 A23 A31 A32 A33. So A11, A12, A13, A21, A22, A23. Second row, third column. favorite second column. So we can write a 23 for finding the minor of the element row and column for finding the minor row and determined to find three shark m3 m. So 1 into 8 8 7 14 8us 14 - answer m2 Okay. First find the minus and cos of all the elements of the determinant 1 - 2 4 2x for 2 2x2 A1 1 A12 A21 A22 So, so M1 for M1 M12 M12. So M21US M2 3 4us 2 Arow number one even number. So A12 A1 3 number. Next a 2 + 1us a 2 + 2 number. So one difference. Okay. Okay. Determined of a given. 1 - 2 4 3 Okay. AJ of adjint of for finding the adjint of a matrix. Transpose transpose factoring. So first transpose adjint. So adjint. This is my adjint of A. Fore. So, so one more question one more find the minus of the determinus M11 M12 M13 M21 M22 M23 2X 2X M11 1 M12 M21 M22 A11 A12 A21 A22 M1 First time in your class. Okay. M1 M2 M2. Okay. No sign change. 1 + 2 mus 2 even number. No sign change. So that is final answer. 0 42. Okay. Transpose first row first. Second row. Second column. This is the final answer. Very good. I should very good. Find adjint for the matrix ALAL to 234. Yes. 2x2. So we can use the trick and we will get full marks. directly of 2x2 only. So, Yeah. Hello. Sus reverse. This is show of egg. S - 4 + I - 2 + 2 I 4 - 4 I 3 2 - 4 1 Okay. Okay. So next 3x3 readyation foreign. Okay. A22 A23 A31 A32 A33 First five into 10us one - 1 - 1 10 -1 random plus I 10 + 11 item. Next 0 into -1 6 m into 1us 5. Next M2 into 1 into 4 - M2 1 into 2 - 0 into 4 2 - 4 2 - So m23 1 into 1 - 0 which means 1 - 1 m3 0 into - 1 0 5 into 4 20 m 1 into -1 -1 3 into 4 12 -1 -2 -3 m33 1 into 5 - 3 into 0 5 - Okay. So A12 A21 A23 A3 number 1 + 2 3 order 2 + 1 3 order 2 + 3 order 3 A1 A22 A31 A3 - 6 A 2 1 + 4 A 23 -1 A32 Okay. So Okay. So first row 11 - 6 3 second row 4 2 -1 third row - 20 13 5in first row second row second third row third column 11 4 - 20 - 2 sorry - 62 13 3 - 5 Okay next properties It's me M1 to 7 M1 to M12 6. So a into joint is equal to adjal into iet of adjint is equal to determin of a allus Determined of determined of a ra 2x2 mat with determined of a determined of adjoin a Okay. of determined. So 5 to 2 - 1 5 1 the answer is it's a important question. Okay, ready? One more question. If A is a 3x3 matrix, then determin the answer is option B. Right answer. Okay. Singular singular determined value singular determined a value notal singular not singular single determinatal non singular not okay inverse inverse Nonveretermin Determination 1 by non-s singular Inverse A inverse A inverse equal to 1 by determination A inverse equation. Find inverse A = 2 - 2 4 Find the inverse of the matrix A inverse equation A inverse 1 determinant of A into signus 2 Plus fourus determined of a determined find 2 into 3us 4 into - 8 - 6 + 8 14 A inverse = 1 by determinant of A 14 into 32us 4. This is my final answer inverse equation. So we moving to the last topic which is application of determinance and matrices application Yet protein. Okay. Very very important. foreign. So equations equations system of equation solving of equations. So first unknowns A1 B1 C1 First row. Second equation. A2 B2 C2. Second row. A3 B3 C3. Third row. Okay. Unknown XY Z unknown matx D1 D2 D3 A into X = matv mat into x = b a coefficient matrix x unknown matal. Okay. Next solving we can write x = a inverse into first question. for Okay, ready into x= equations. X unknown X into X= first Solving we need x x= equation a inverse into a inverse inverse 1 by determin of a into adjint of a inverse. So determined of a so determined of a So 5 into 3 - 7 into 2 5 into 3 15 - 14 is equal to ital joint of a joint of a 2x 7. So a inverse = 1 3 - 7 1 by 1 1 by 1 1. So we can write a inverse = 3 - 2 - 7 5 = a inverse inver 3 - 2 - 7 into element five mat First column 3 into 4 12 - 2 into 5 - 10 second row into second column - 7 into 4 - 28 5 into 5 12US 10 - 28 + 25 italus then x = y= Right answer. [Music] Then we moving to the last. This is our last question. Last question. Consider system of equations. Moon equations. But 2x + 3 y + 3 z= 5. Second equation x - 2 y + z= - 4. Third one 3x - 3x - y - 2 = 3. Okay. Write it format for a into x = 3. Second equation 1 - 2 1 Third 3 - 1 - 2 3 3 1 - 2 1 3 -1 - 2 X unknown X Y Okay. Mat prove that the system is consistent. System of equations of consistent determined of Determined of a plus -1us. So 4 - 1 into - 2 - 2 3 3 into 1 3. So - last 1 into -1 -1 3 into -2 - 6. So -1 - 6 -1 - - - - - - - - - - - - - - - - - - - - - 6 2 into 4 + 1 5 - 3 into - 2 - 3 - 5 + 3 into -1 + 5 of a note. equal to singularist consistent. Okay. So we moving to the question solve system of equation using method. solve. So we know that x = a inverse. So a inverse 1 by determined. So Mr. M11, M12, M13, M21, M22, M23, M31, M32, M33. So mus 1 -1 4 - -1 5 m - 2 into 1 - 3 into 1 - 2 - 3 - 5 M1 -1 3 into - 2 - 6 -1 - - 6 -1 + 6 5. Okay. M2 3 into -2 - 6 -1 3 - 3 - 6 - -3 - 6 + 3 M2 into -4 3 into 3 9 - 4 - 9 it is -3 M3 M2 into -1 - 2 3 3 into 3 9 - 2 - 9 - 11 M3 3 into 1 3 - 2 into 3 - 6 3 - - 6 3 + 6 9 simple 3 into 1 2 - 3 it is -1 last 2 into -2 -4 3 into 1 3 -4 - 3 - the speed. Okay. A11 A12 A13 A21 A22 A23 A31 A32 A331 A1 A1 A3 A3 So five a fiveus 2 three + 11 + final row 55 second row 3 - 13 11 third row 9 1us row second row second row finished A inverse last 1 of A into 1 by 40 into Yes. Moving to the last. A inverse guys 1 by 14 B mat 1 by matiplic good boy. So first row into first column 5 into 5 25 3 into - 4 - 12 9 into 3 27 second row into first column 50 the three third row into first column 25 - 44us 3 25 - 12 + 2 it is 40 Yes. 25 + 52 + 3 80 25 - 3 - 44 3 Yes. 25 - 21 - 44 it is - 40 1x 40 40 into 1 by 40 80 into 1x 40 - 40 into 1. So minus one. So the answer is unknown. X1 Y2us one. Finished. last because so So, thank you all for watching. So next class. So bye-bye all. Thank you. Love you all. Bye. Thank you. Thank you.