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Chapter 7: Q27E (page 383)

If an invertible matrix A is diagonalizable, then A-1 must be diagonalizable as well.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define eigenvalue

Assume that 位is eigenvalue of A which is an invertible and diagonalizable matrix, then 1/位 must be the eigen value of A-1, so that it will also be diagonalizable.

02

Explanation

If an invertible matrix A is diagonalizable, then there exists an invertible matrix S and a diagonal matrix D such that

S-1AS=D

From this, we get

S-1A-1(S-1)=D-1

S-1AS=D

Since D-1 is also diagonal, we conclude that A-1 is also diagonalizable.

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Most popular questions from this chapter

Consider a 4 脳 4 matrixA=|BC0D|where B, C, and D are 2 脳 2 matrices. What is the relationship among the eigenvalues of A, B, C, and D?

suppose a certain4脳4 matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(e鈫�1) and span(e鈫�1,e鈫�2) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (e鈫�1) and span (e鈫�1,e鈫�2) are A-invariant subspaces of R3 . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Orthogonal projection onto a line L inR3.

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Orthogonal projection onto a line L in R3.
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