To prove: If A is an invertible matrix and k is a positive integer, then ( A k ) − 1 = A − 1 A − 1 ⋅ ⋅ ⋅ A − 1 ︸ k − f a c t o r s = ( A − 1 ) k .

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Answer 1

Question

Chapter A, Problem 11E

To determine

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

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Problem 1 & 2 answers 1. One diagonal has 11 squares, then total square in total for two diagonal line is 11 + 11 - 1 = 21 . 2. Each part has 5 squares.(except middle)Multiply by 4: 5 × 4 = 20.Add the middle square: 20 + 1 = 21.

2. Now Figure out a different way you could determine how many squares there are in the figure, again without counting them all one-by-one. Briefly describe this other method:

1. Without counting all of the squares one by one, determine how many squares there are in the figure shown. Briefly describe your method.

Answer 2

Question

Chapter A, Problem 11E

To determine

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

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Answer 3

Question

Chapter A, Problem 11E

To determine

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

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Answer 4

Question

Chapter A, Problem 11E

To determine

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

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Answer 5

Chapter A, Problem 11E

To determine

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

Answer 6

Chapter A, Problem 11E

To determine

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

Answer 7

Chapter A, Problem 11E

To determine

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. A - Using Mathematical Induction In Exercises 1-4, use...Ch. A - Prob. 2E Ch. A - Prob. 3E Ch. A - Prob. 4E Ch. A - Prob. 5E Ch. A - Prob. 6E Ch. A - Prob. 7E Ch. A - Prob. 8E Ch. A - Prob. 9E Ch. A - Prob. 10E

To prove: If A is an invertible matrix and k is a positive integer, then ( A k ) − 1 = A − 1 A − 1 ⋅ ⋅ ⋅ A − 1 ︸ k − f a c t o r s = ( A − 1 ) k .

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To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

Want to see the full answer?

Chapter A Solutions

To prove: If A is an invertible matrix and k is a positive integer, then ( A k ) − 1 = A − 1 A − 1 ⋅ ⋅ ⋅ A − 1 ︸ k − f a c t o r s = ( A − 1 ) k .

Videos

To prove: If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.

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Chapter A Solutions

To prove:

If A is an invertible matrix and k is a positive integer, then (Ak)−1=A−1A−1⋅⋅⋅A−1︸k−factors=(A−1)k.