Find All Symmetric Matrices satisfying the Equation
Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix}
1 \\
-1
\end{bmatrix}
=
\begin{bmatrix}
2 \\
3
\end{bmatrix}$? Express your solution using free variable(s).
Solution.
Let $A=\begin{bmatrix}
a & b\\
c& d
\end{bmatrix}$ […]

The Rank of the Sum of Two Matrices
Let $A$ and $B$ be $m\times n$ matrices.
Prove that
\[\rk(A+B) \leq \rk(A)+\rk(B).\]
Proof.
Let
\[A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],\]
where $\mathbf{a}_i$ and $\mathbf{b}_i$ are column vectors of $A$ and $B$, […]

Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
(a) Let $A=\begin{bmatrix}
1 & 3 & 0 & 0 \\
1 &3 & 1 & 2 \\
1 & 3 & 1 & 2
\end{bmatrix}$.
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.
(b) Find the rank and nullity of the matrix $A$ in part (a).
Solution.
(a) […]

A Group Homomorphism and an Abelian Group
Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$.
Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.
Proof.
$(\implies)$ If $G$ is an abelian group, then $f$ […]

Determine a 2-Digit Number Satisfying Two Conditions
A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45.
Find the number.
Solution.
The key to this problem is noticing that our 2-digit number can be […]

All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8
Determine all the conjugacy classes of the dihedral group
\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\]
of order $8$.
Hint.
You may directly compute the conjugates of each element
but we are going to use the following theorem to simplify the […]

Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$
Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.
(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.
(b) Let
\[A^{100}=aA^2+bA+cI,\]
where $I$ is the $3\times 3$ identity matrix.
Using the […]