Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q 73E (page 101)

Question: For two invertible n 脳 n matrices Aand B , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.
${(A B A^{鈭� 1})}^{3} = A B^{3} A^{鈭� 1}$

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:

Any matricesA andB that do not commute because the matrices are of order n 脳 n.

In general commutative law for multiplication is not exist that is $A B 鈮� B A$ .

Determine the given equation of matrices is true or false.

Solve the left hand side of the given equation gives:

$\begin{matrix} {(A B A^{鈭� 1})}^{3} = (A B A^{鈭� 1}) (A B A^{鈭� 1}) (A B A^{鈭� 1}) \\ = A B (A^{鈭� 1}) (A) B (A^{鈭� 1}) (A) B A^{鈭� 1} \\ = A B (I) B (I) B A^{鈭� 1} \end{matrix}$

Solve further as:

$\begin{matrix} {(A B A^{鈭� 1})}^{3} = A B (B) (B) A^{鈭� 1} \\ = A B^{3} A^{鈭� 1} \end{matrix}$

Then it gives ${(A B A^{鈭� 1})}^{3} = A B^{3} A^{鈭� 1}$ .

Hence, the given equation is true.

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91影视

Question: For two invertible n 脳 n matrices Aand B , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.
${(A B A^{鈭� 1})}^{3} = A B^{3} A^{鈭� 1}$

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.

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91影视

Question: For two invertible n 脳 n matrices Aand B , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.(ABA鈭�1)3=AB3A鈭�1

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

Calculus

Mechanics Maths

Statistics

Geometry

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Question: For two invertible n 脳 n matrices Aand B , determine which of the formulas stated in Exercise 67 through 75 are necessarily true.
${(A B A^{鈭� 1})}^{3} = A B^{3} A^{鈭� 1}$