. Matrices A and B are said to be similar if and only if there is a matrix P such that A = PBP^-1 . If A and B are similar, show that 1) Det A = det B 2) A and B together have an inverse or do not have an inverse. 3) The characteristic equations of A and B are the same.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 11AEXP
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13. Matrices A and B are said to be similar if and only if there is a matrix P such that A = PBP^-1 . If A and B are similar, show that

1) Det A = det B

2) A and B together have an inverse or do not have an inverse.

3) The characteristic equations of A and B are the same.

4) The eigenvalues ​​of A are the same as the eigenvalues ​​of B.

5) The eigenvectors A and B can be different.

Please solve this problem complete please no reject

 

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