# Calculate $A^{10}$ for a Given Matrix $A$

## Problem 41

Find $A^{10}$, where $A=\begin{bmatrix} 4 & 3 & 0 & 0 \\ 3 &-4 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$.

(Harvard University Exam)

## Solution.

Let $B=\begin{bmatrix} 4 & 3\\ 3& -4 \end{bmatrix}$ and $C=\begin{bmatrix} 1 & 1\\ 1& 1 \end{bmatrix}$ and write $A=\begin{bmatrix} B & 0\\ 0& C \end{bmatrix}$ as a block matrix.
Then we have
$A^{10}=\begin{bmatrix} B^{10} & 0\\ 0& C^{10} \end{bmatrix}.$

It remains to calculate $B^{10}$ and $C^{10}$.

We have $B^2=5^2I$, where $I$ is the $2\times 2$ identity matrix. From this, we get $B^{10}=5^{10}I$.

Also, we have $C^2=2C$. Applying this repeatedly, we get $C^{10}=2^9C$.

Therefore we have
$A^{10}=\begin{bmatrix} 5^{10} & 0 & 0 & 0 \\ 0 &5^{10} & 0 & 0 \\ 0 & 0 & 2^9 & 2^9 \\ 0 & 0 & 2^9 & 2^9 \end{bmatrix}.$

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