(1) We prove $A^n=\begin{bmatrix}
a^n & 0\\
0& b^n
\end{bmatrix}$ by induction on $n$.
The base case $n=1$ is true by definition.

Suppose that $A^k=\begin{bmatrix}
a^k & 0\\
0& b^k
\end{bmatrix}$. Then we have
\[A^{k+1}=AA^k =\begin{bmatrix}
a & 0\\
0& b
\end{bmatrix}
\begin{bmatrix}
a^k & 0\\
0& b^k
\end{bmatrix}
=\begin{bmatrix}
a^{k+1} & 0\\
0& b^{k+1}
\end{bmatrix}.\]
Here we used the induction hypothesis in the second equality.
Hence the inductive step holds. This completes the proof.

(2) We show that $B^n=S^{-1}A^n S$ by induction on $n$.
When $n=1$, this is just the definition of $B$.

For induction step, assume that $B^k=S^{-1} A^k S$.
Then we have
\[B^{k+1}=B B^k=(S^{-1} A S) (S^{-1} A^k S)=S^{-1}A A^k S=S^{-1} A^{k+1} S,\]
where we used the induction hypothesis in the second equality and the third equality follows by canceling $S S^{-1}=I_2$ in the middle.

Thus the inductive step holds, and this competes the proof.

The Powers of the Matrix with Cosine and Sine Functions
Prove the following identity for any positive integer $n$.
\[\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta
\end{bmatrix}^n=\begin{bmatrix}
\cos n\theta & -\sin n\theta\\
\sin n\theta& \cos […]

Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector
Let
\[A=\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}
1\\
0
\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.
(The Ohio State University, Linear Algebra Exam)
Solution.
We first compute $A\mathbf{u}$. We […]

Problems and Solutions About Similar Matrices
Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix.
Prove the following statements.
(a) If $A$ is similar to $B$, then $B$ is similar to $A$.
(b) $A$ is similar to itself.
(c) If $A$ is similar to $B$ and $B$ […]

A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Definitions/Hint.
Recall the relevant definitions.
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]

Dimension of Null Spaces of Similar Matrices are the Same
Suppose that $n\times n$ matrices $A$ and $B$ are similar.
Then show that the nullity of $A$ is equal to the nullity of $B$.
In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]

Find the Formula for the Power of a Matrix
Let
\[A=\begin{bmatrix}
1 & 1 & 1 \\
0 &0 &1 \\
0 & 0 & 1
\end{bmatrix}\]
be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.
Proof.
We first compute several powers of $A$ and guess the general formula.
We […]

Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$
Let $A$ be the coefficient matrix of the system of linear equations
\begin{align*}
-x_1-2x_2&=1\\
2x_1+3x_2&=-1.
\end{align*}
(a) Solve the system by finding the inverse matrix $A^{-1}$.
(b) Let $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ be the solution […]

Companion Matrix for a Polynomial
Consider a polynomial
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.
Define the matrix
\[A=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & […]

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