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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

Answer

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

Answer

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

Answer

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

Answer

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.

Answer

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

Answer

Exercise Prove that

[ a b ]
c d

is invertible ⇔ ad - bc ≠ 0.

Answer

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

Answer

Exercise Let A and B be square matrices. Prove that A and B are invertible ⇔ AB is invertible. Please also explain why the square condition is necessary.

Answer

Exercise

  1. Prove that Ax = b has solutions for any b ⇔ There is a matrix X, such that AX = I.
  2. Prove that solution of Ax = b is unique ⇔ There is a matrix X, such that XA = I.
  3. Rephrase the first and second parts in terms of linear transformations.

Answer