Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
(a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\]
where […]

Similar Matrices Have the Same Eigenvalues
Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.
Proof.
We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]

Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$
Let $T:\R^3 \to \R^2$ be a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
0
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
0 \\
1
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
1 \\
0
\end{bmatrix},\]
where $\mathbf{e}_1, […]

Linear Transformation $T:\R^2 \to \R^2$ Given in Figure
Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below.
Find the matrix representation $A$ of the linear transformation $T$.
Solution 1.
From the figure, we see […]

Sequence Converges to the Largest Eigenvalue of a Matrix
Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.
Furthermore, suppose that
\[|\lambda_1| > |\lambda_2| \geq \cdots \geq […]