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 $C^{10}=2^9C$.

Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)
(a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.
Find $\det(A)$.
(b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
(c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]

Determinant of Matrix whose Diagonal Entries are 6 and 2 Elsewhere
Find the determinant of the following matrix
\[A=\begin{bmatrix}
6 & 2 & 2 & 2 &2 \\
2 & 6 & 2 & 2 & 2 \\
2 & 2 & 6 & 2 & 2 \\
2 & 2 & 2 & 6 & 2 \\
2 & 2 & 2 & 2 & 6
\end{bmatrix}.\]
(Harvard University, Linear Algebra Exam […]

Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix.
Let A be the matrix
\[\begin{bmatrix}
1 & -1 & 0 \\
0 &1 &-1 \\
0 & 0 & 1
\end{bmatrix}.\]
Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse.
(The Ohio State University Linear Algebra […]

Find the Rank of a Matrix with a Parameter
Find the rank of the following real matrix.
\[ \begin{bmatrix}
a & 1 & 2 \\
1 &1 &1 \\
-1 & 1 & 1-a
\end{bmatrix},\]
where $a$ is a real number.
(Kyoto University, Linear Algebra Exam)
Solution.
The rank is the number of nonzero rows of a […]

Find the Inverse Matrix of a $3\times 3$ Matrix if Exists
Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 1 & 2 \\
0 &0 &1 \\
1 & 0 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(The Ohio State University, Linear Algebra Midterm Exam […]

Idempotent Matrices. 2007 University of Tokyo Entrance Exam Problem
For a real number $a$, consider $2\times 2$ matrices $A, P, Q$ satisfying the following five conditions.
$A=aP+(a+1)Q$
$P^2=P$
$Q^2=Q$
$PQ=O$
$QP=O$,
where $O$ is the $2\times 2$ zero matrix.
Then do the following problems.
(a) Prove that […]