## How to Find a Formula of the Power of a Matrix

## Problem 8

Let $A= \begin{bmatrix}

1 & 2\\

2& 1

\end{bmatrix}$.

Compute $A^n$ for any $n \in \N$.

Let $A= \begin{bmatrix}

1 & 2\\

2& 1

\end{bmatrix}$.

Compute $A^n$ for any $n \in \N$.

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 $n \in \N$

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*)

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

Add to solve laterA square matrix $A$ is called **idempotent** if $A^2=A$.

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

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