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

Math exam problems and solutions at Harvard University

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)

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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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