## True of False Problems on Determinants and Invertible Matrices

## Problem 438

Determine whether each of the following statements is True or False.

**(a)** If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

**(b)** If the characteristic polynomial of an $n \times n$ matrix $A$ is

\[p(\lambda)=(\lambda-1)^n+2,\]
then $A$ is invertible.

**(c)** If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

**(d)** If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

**(e)** If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(*Stanford University, Linear Algebra Exam Problem*)

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