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Chapter 2: Q2E (page 107)

If A is any invertiblen脳n matrix, then A commutes with A-1.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

By Theorem 2.4.6

From Theorem2.4.6, (Multiplying with the inverse), we know that

For an n脳ninvertible matrix A,

A-1A=In=AA-1

02

Check for the given statement

It is given that the given matrix A is invertible, then by using theorem 2.4.6, A and A-1commute with each other.

Hence, the given statement is true.

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Most popular questions from this chapter

Some parking meters in downtown Geneva, Switzerland, accept2Franc and5 Franc coins.

a. A parking officer collects 51 coins worth 144Francs. How many coins are there of each kind?

b. Find the matrixAthat transforms the vector

[number of 2 Franc coinsnumber of5Franc coins]

into the vector

[total value of coinstotalnumberof coins]

c. Is the matrixAin part (b) invertible? If so, find the inverse (use Exercise 13). Use the result to check your answer in part (a).

If A is a 3脳4matrix and B is a4脳5, then AB will be a5脳3matrix.

Suppose A is a transition matrix and B is a positive transition matrix (see Definition 2.3.10), where A andB are of the same size. Is AB necessarily a positive transition matrix? What about BA?

If matrices A and B commute, then the formula A2B = BA2 must hold.

Find all 2 脳 2 matrices X such that AX = X A for all 2 脳 2 matrices A.
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