### Inverse

Exercise Find invertible matrices and compute the inverse.

 [ 3 -2 ] 5 -3
 [ 3 5 ] -2 -3
 [ 6 4 ] 9 6
 [ 6 4 ] 9 7
 [ 2 0 1 ] 1 1 0 0 1 1
 [ 2 2 1 ] 1 -1 0 0 -2 1
 [ 2 0 2 ] 1 1 -1 0 1 -2
 [ 2 0 2 1 1 ] 1 1 -1 0 -1 0 1 -2 1 -2
 [ 1 2 1 1 ] 2 1 1 0 0 3 1 0 3 0 1 0
 [ 1 2 3 4 ] 0 1 2 3 0 0 1 2 0 0 0 1
 [ 10.2 0.3 -5.4 ] -7.9 8.2 12.4 0.8 -0.9 1.6 33.5 -44.9 38.4 -5.9 -7.3 0.1
 [ 1 1 1 1 1 ] 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0

Exercise Find invertible linear transformations and compute the inverse.

T1(x1, x2) = (2x1 + 3x2, 5x1 + 7x2): R2R2

T2(x, y) = (2x - 6y, - 5x + 15y): R2R2

T3(x1, x2, x3) = (- 2x2 + x3, x1 - x3, 3x1 + x2 - 3x3): R3R3

T4(x1, x2, x3) = (- 2x2 + x3, 3x1 + x2 - 3x3): R3R2

T5(x1, x2, x3) = (- 2x2 + x3, x1 - x3, 3x1 + 2x2 - 4x3): R3R3

T6(x1, x2, x3, x4) = (x2 + 2x3 + 4x3, x1 + x2 + 2x3, x1 + x2 + 2x3 - 2x4, x1 + x2 + x3 + 2x4): R4R4

T7(x1, x2, x3, x4, x5) = (x1 + x2, x2 + x3, x3 + x4, x4 + x5, x5 + x1): R5R5

Exercise For what choices of the parameters, the following matrices are invertible?

 [ a + 1 a - 1 ] a a + 2
 [ 1 1 1 1 ] 1 0 0 0 2 2 0 a 0 1 3 a
 [ 1 1 -1 ] 1 0 -a 1 a 0
 [ 0 1 1 a ] 1 0 a 1 1 1 0 0 0 a -1 1
 [ 1 0 0 0 ] a 1 0 0 0 a 1 0 0 0 a 1
 [ 1 1 1 1 ] a 1 1 1 a a 1 1 a a a 1
 [ a b b b ] a a b b a a a b a a a a

Exercise Compute the inverse of the following matrices.

 [ 2 3 ] 7 9
 [ 1 0 2 ] 2 -1 3 4 -1 8
 [ 2 0 2 1 ] 1 1 -1 0 0 1 -2 -1 1 -1 2 1

Then use your result to solve the following systems.

 2x1 + 3x2 = 2 7x1 + 9x2 = -1
 2x1 + 3x2 = 1 7x1 + 9x2 = 0
 x1 + 2x3 = 1 2x1 - x2 + 3x3 = 0 4x1 - x2 + 8x3 = -1
 x1 + 2x3 = 0 2x1 - x2 + 3x3 = 2 4x1 - x2 + 8x3 = 1
 2x1 + 2x3 + x4 = 1 x1 + x2 - x3 = 0 x2 - 2x3 - x4 = 1 x1 - x2 + 2x3 + x4 = 0

Exercise Show that

 A = [ 1 0 -2 ], B = [ -3 -2 2 ] 0 1 0 0 1 0 2 1 -3 -2 -1 1

are inverse to each other. Then compute

 C = A[ 1 0 0 ]B 0 -1 0 0 0 0

and C2001, (AC)-1, (A2)-1.

Exercise What is the condition for a diagonal matrix to be invertible? What is the inverse? What about an upper (lower) triangular matrix?

Exercise Prove that

 [ a b ] c d

is invertible ⇔ ad - bc ≠ 0.

Exercise Prove that if row operations change [A B] to [I X], then X = A-1B.