# Category: Linear Algebra

## Problem 8

Let $A= \begin{bmatrix} 1 & 2\\ 2& 1 \end{bmatrix}$.
Compute $A^n$ for any $n \in \N$.

## Problem 7

Let $A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$.
Show that

(1) $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ for any $n \in \N$.

(2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix.
Show that $B^n=S^{-1}A^n S$ for any $n \in \N$

## Problem 5

Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.

(The Ohio State University Linear Algebra Exam)

## Problem 2

Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.

## Problem 1

A square matrix $A$ is called idempotent if $A^2=A$.

Show that a square invertible idempotent matrix is the identity matrix.