Problem #1: Find the values of a, b, and so that det(A) = ax? + bx +c. x -2 0 A=1|-2 x 2 3 -6 -4 Problem #1: : separate your answers with a comma | Just Save ' | Submit Problem #1 for Grading Problem #1 | Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #2: (a) Find the determinant of the following matrix. 3 0 2 0 2 3 0 -3 A=11 0 4 3 1 0 -3 0 1 2 0 0 O -1 2 0 0 o0 B=|3 -2 -2 0 0 3 2 1 2 0 -2 1 2 3 -2 Hint: Do a row operation first. e Y | Just Save ' | Submit Problem #2 for Grading | Problem #2 | Attempt #1 Attempt #2 Attempt #3 Your Answer: | 2(a) 2(a) 2(a) 2(b) 2(b) 2(b) Your Mark: | 2(a) 2(a) 2(a) 2(b) 2(b) 2(b) Problem #3: Consider the following matrix. a b A= d e f g h i Suppose that det(A) = 2. Let B be another 3 x 3 matrix (not given here) with det(B) = 3. Find the determinant of each of the following matrices. (a) a+2g -g -2d C=|b+2n -h =2e c+2i -i =2f (b) D=24"1B8T (c) E=-BA? (d) F=adj(B) Problem #3(a): Problem #3(b): Problem #3(c): Problem #3(d): as in these examples | Just Save | | Submit Problem #3 for Grading | Problem #3 | Attempt #1 Attempt #2 Attempt #3 Your Answer: | 3(a) 3(a) 3(a) 3(b) 3(b) 3(b) 3(c) 3(c) 3(c) 3(d) 3(d) 3(d) Your Mark: | 3(a) 3(a) 3(a) 3(b) 3(b) 3(b) 3(c) 3(c) 3(c) 3(d) 3(d) 3(d) Problem #4: Consider the following matrix. -8 -1 -5 A=| 2 1 -3 4 4 1 Let B = adj(A). Find b31, b3y, and b33. (i.e., find the entries in the third row of the adjoint of A.) Problem #4: I:I separate your answers with a comma | Just Save | | Submit Problem #4 for Grading Problem #4 Your Answer: Your Mark: Attempt #1 Attempt #2 Attempt #3 Problem #5: Let A and B be n x n matrices. Which of the following statements are always true? (i) If det(A) = det(B) then det(A B) =0. (ii) If A and B are symmetric, then the matrix AB is also symmetric. (iii) If A and B are skew-symmetric, then the matrix A7 + B is also skew-symmetric. (A) none of them (B) (i) and (ii) only (C) (i) and (iii) only (D) (iii) only (E) all of them (F) (ii) only (G) (i) only (H) (ii) and (iii) only Problem #5: | Just Save ' ' Submit Problem #5 for Grading Problem #5 | Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #6: Find the eigenvalues of the following matrix. 4 -6 4 A=|0 -5 0 2 -6 -2 Problem #6: I:I separate your answers with a comma | Just Save ' ' Submit Problem #6 for Grading Problem #6 | Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #7: Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x. -7 -8 -8 2 A= | * * * | x=|-6 * * *# -8 Given that x is an eigenvector of the matrix A, what is the corresponding eigenvalue? | Just Save ' ' Submit Problem #7 for Grading Problem #7 | Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem 1: finding the deteriwat Problems. finding the determinat of metrix A niven matrix A: X - 2 0 A - 2 2 - 61- 4 are tasked with finding det LAJ - detin ) - and Font c solution using cofactor expansion along the first you : " 2 (z todel 2 2 10 det (A) = n der - 6 - 4 - 3- 4 Now, calculate the determination of the 242 matrices, det x 2 n. (- 4 ) - 2 . 1 - 6 ) 2 42 +12 - 4 2 2 2- 2. (- 4) - 2-(- 37 - 8+6 2 14 Now. substitute back into the determment formally det (A) = x . ( - 4 2 + 12 ) + 2-14simplifying det ( A ) = " In2 + 122+ 26 + of Thus, comparing this with antbite we find a = - 4 - 6= 12 , C = 26 Problem 2. - LAJ = part ( a) : metrix A -3 O 2 0 A = 2 - 3 - 3 4 3 we first - 2 2 10 Exland along the second column( since it - 3 - 4 has zenos ) ! 2 f the det ( A ) = - ( -3 1 , det 4 - 3 0 Now calculate the determination of 3x3 methy diet 1.2 0 - 3 214 L 4 3 - 3 0 using cofactor expansion along the first emment 5 = 2. (4+0 - 3. 1- 31) = 2.9 21 dot = 2. det therefore dot (A) = 3.18 = 54Part(b ) - matrix 2 - I 2 B = - 2 - 2 2 2 0 L 2 3 - 2 perform a row ofretion 2 O '0 2 -21- 2 3 2 - 2 2 3 - 2 since the second rove is all zero. det ( 1) ) = 0 and tweefore bet ( B) 20 . det ( B ) =0 . Problems( s] - ( 3 ) determinants of various metrics let ( A ) = - 2 1 det ( B ) = 3 part(a) : matrix since matrix cis obtained by adding multiple of the third column to the first and second columns determinants remains unchanged: "det ( c ) = det ( A) = - 21part (b) - matrix ( D ) to AShivefar wing the formally dot (kaj = kh, det (a ) for an nyn matrix . dot ( D ) = diet ( 2At B ) = 23. let (AT ) . det ( ? ) since dot ( Al ) Jet (A ) and def (BT ) = dot ( B ). we have det ( D ) = Bit . 3 2 - 12 Pant (c ) matrice using the property diet ( AB) = dot (A) . det(B) det (E ) = dot ( - BAZ ) = 1 -1 1 3. det ( B ) . det ( A ) ? det (E) = = 1.3.1-212 = -1: 3, 45 - 12 (B 7 20 . Portfor matrix -A ( Adpoint of B ) -The determinant of the adjoint of a given by : ( det ( B ) n - 1 det ( adj ( B ) ) - we have for 9 3 x 3 matrix det ( p ) = (dot ( 1 ) ) 2 2 3 2 9Shuvatar Problem 4 - finding Entries in the Third row of the ddjoint of A - B - 1 - 5 AZ 4 4 we are tasked with finding 671 632 . biz The entries in two third of the adJoint of A To find the adjoint calculate the Cofactors for the third rowe ! is Cafactor of Q21 : - 21 bill = [ - 1 ) . def + , = 1 . ( 1+ 12 ) 2 13 8 632 is cafactor of 9321. 3+2 1 632 = (-1) -5 1 det -2 7- 1. (24 - 10 ) 2 - 14 633 is the cofactor of ass, 633 = (- 1 )375 - def = 1. (- B+ 2 ) = - 6 2 They $31 2 13 , 632 2 -14 , 629 = - 6Problem-6 - finding the eigenvalues of metrix At 4 - 6 4 - 6 - 2 4 -2 solve the characteristics equation dot ( A - XT ) SO Expanding twig dot ( A - AT ) = 13 - 47 - 8 x + 69 The eigenvalues are approximately . 1 1 = 6 17 24 2 3 2- 4 Problem 7 - finding the cigenially of a Block diagonal matrix for a block diagonal matrix the eigenvalue are simple the eigenvalues of the diagonal blocks . since A is a 2 x 2 block and 13 is 9 3x3 block * ( A ) = 5- 1:34, 2 ( B ) = $ 1 1 0.42 They the rigonvalue of My are 5-1, 31 1 10,4 4