430 VECTORS AND MATRICES [APP. A
A.30. LetA=[
1 − 12
034]
,B=[
40 − 3
− 1 − 23]
,C=⎡
⎣2 − 301
5 − 1 − 42
− 1003⎤
⎦,D=⎡
⎣2
− 1
3⎤
⎦.Find: (a) 3A− 4 B; (b)AB,AC,AD; (c)BC,BD,CD; (d)ATandATB.A.31. LetA=[
12
36]
.Find a 2×3 matrixBwith distinct entries such thatAB=0.SQUARE MATRICES
A.32. Find the diagonal and trace of: (a)A=⎡
⎣2 − 78
3 − 6 − 5
40 − 1⎤
⎦; (b)B=⎡
⎣12 − 9
− 328
5 − 6 − 1⎤
⎦.A.33. LetA=[
2 − 5
31]
.Find: (a)A^2 andA^3 ; (b)f (A)wheref(x)=x^3 − 2 x^2 −5.A.34. LetB=[
4 − 2
1 − 6]
.Find: (a)B^2 andB^3 ; (b)f(B)wheref(x)=x^2 + 2 x−22.A.35. LetA=[
6 − 4
3 − 2]
.Find a nonzero vectoru=[
x
y]
such thatAu= 4 u.DETERMINANTS AND INVERSES
A.36. Find each determinant: (a)∣∣
∣∣^25
41∣∣
∣∣;(b)∣∣
∣∣^61
3 − 2∣∣
∣∣;(c)∣∣
∣∣−^28
− 5 − 2∣∣
∣∣;(d)∣∣
∣∣a−ba
aa+b∣∣
∣∣.A.37. Compute the determinant of each matrix in Problem A.32.A.38. Find the inverse of: (a)A=[
74
53]
;(b)B=[
5 − 2
6 − 3]
;(c)C=[
4 − 6
− 23]
.A.39. Find the inverse of each matrix (if it exists):A=⎡
⎣12 − 4
− 1 − 15
27 − 3⎤
⎦; B=⎡
⎣1 − 11
02 − 2
13 − 1⎤
⎦; C=⎡
⎣123
25 − 1
512 1⎤
⎦.ECHELON MATRICES, ROW REDUCIONS, GAUSSIAN ELIMINATION
A.40. ReduceAto echelon form and then to row canonical form, where:(a)A=⎡
⎣12 − 121
24 1− 23
36 2− 65⎤
⎦; (b)A=⎡
⎣23 − 251
3 − 1204
4 − 56 − 57⎤
⎦.A.41. Using only 0’s and 1’s, list all 2×2 matrices in echelon form.
A.42. Using only 0’s and 1’s find the number of 3×3 matrices in row canonical form.
A.43. Solve each system:(a)x+ 2 y− 4 z=− 3
2 x+ 6 y− 5 z= 2
3 x+ 11 y− 4 z= 12(b)x+ 2 y− 4 z= 3
2 x+ 6 y− 5 z= 10
3 x+ 10 y− 6 z= 14A.44. Solveeachsystem:(a)x− 3 y+ 2 z− t= 2
3 x− 9 y+ 7 z− t= 7
2 x− 6 y+ 7 z+ 4 t= 7(b)x+ 2 y+ 3 z= 7
x+ 3 y+ z= 6
2 x+ 6 y+ 5 z= 15
3 x+ 10 y+ 7 z= 23