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 $n \in \N$.
Add to solve later Category: Linear Algebra
Add to solve later Powers of a Diagonal Matrix
Problem 7
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 $n \in \N$
Add to solve later A Linear Transformation Maps the Zero Vector to the Zero Vector
Problem 5
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)
Add to solve later Similar Matrices Have the Same Eigenvalues
Problem 2
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 later Invertible Idempotent Matrix is the Identity Matrix
Problem 1
A 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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