Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q 70E (page 100)

Question: For two invertible n 脳 n matricesA andB , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.
70.A² is invertible, and ${(A^{2})}^{鈭� 1} = {(A^{鈭� 1})}^{2}$

Short Answer

Expert verified

Thus, the given equation is true.

Step by step solution

Use associative laws for the given matrices.

The given matrices areA andB which is invertible and solve the given equation as follows:

By definition of matrix inverse, ${(A A)}^{鈭� 1} (A A) = I$ .

Now, associative property of products of matrices written as follows:

$({(A A)}^{- 1} A) A = I$

Determine the given equation of matrices is true or false.

Post-multiplication of A^-1in the equation $({(A A)}^{鈭� 1} A) A = I$ gives:

$\begin{array}{l} ({(A A)}^{鈭� 1} A) A = I \\ ({(A A)}^{鈭� 1} A) A A^{鈭� 1} = A^{鈭� 1} \\ {(A A)}^{鈭� 1} A = A^{鈭� 1} \end{array}$

Again, Post-multiplication of A^-1 in the above equation gives;

role="math" localid="1664190604978" $\begin{array}{l} {(A A)}^{鈭� 1} A A^{鈭� 1} = A^{鈭� 1} A^{鈭� 1} \\ {(A A)}^{鈭� 1} = A^{鈭� 1} A^{鈭� 1} \\ {(A^{2})}^{鈭� 1} = {(A^{鈭� 1})}^{2} \end{array}$

Then it gives ${(A^{2})}^{鈭� 1} = {(A^{鈭� 1})}^{2}$ .

Thus,A² has an inverse if and only ifA has a inverse.

So, A² is an invertible and ${(A^{2})}^{鈭� 1} = {(A^{鈭� 1})}^{2}$

Hence, the given equation is true.

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91影视

Question: For two invertible n 脳 n matricesA andB , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.
70.A² is invertible, and ${(A^{2})}^{鈭� 1} = {(A^{鈭� 1})}^{2}$

Short Answer

Step by step solution

Use associative laws for the given matrices.

Determine the given equation of matrices is true or false.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

Question: For two invertible n 脳 n matricesA andB , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.70.A2 is invertible, and(A2)鈭�1=(A鈭�1)2

Short Answer

Step by step solution

Use associative laws for the given matrices.

Determine the given equation of matrices is true or false.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Decision Maths

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Question: For two invertible n 脳 n matricesA andB , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.
70.A² is invertible, and ${(A^{2})}^{鈭� 1} = {(A^{鈭� 1})}^{2}$