## Problems and Solutions About Similar Matrices

## Problem 319

Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix.

Prove the following statements.

**(a)** If $A$ is similar to $B$, then $B$ is similar to $A$.

**(b)** $A$ is similar to itself.

**(c)** If $A$ is similar to $B$ and $B$ is similar to $C$, then $A$ is similar to $C$.

**(d)** If $A$ is similar to the identity matrix $I$, then $A=I$.

**(e)** If $A$ or $B$ is nonsingular, then $AB$ is similar to $BA$.

**(f)** If $A$ is similar to $B$, then $A^k$ is similar to $B^k$ for any positive integer $k$.