An Example of a Real Matrix that Does Not Have Real Eigenvalues
Let
\[A=\begin{bmatrix}
a & b\\
-b& a
\end{bmatrix}\]
be a $2\times 2$ matrix, where $a, b$ are real numbers.
Suppose that $b\neq 0$.
Prove that the matrix $A$ does not have real eigenvalues.
Proof.
Let $\lambda$ be an arbitrary eigenvalue of […]

$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.
Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.
Proof.
Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$.
Define the function $\Psi:R[x,y] \to R[t]$ sending […]

Unit Vectors and Idempotent Matrices
A square matrix $A$ is called idempotent if $A^2=A$.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]

Determine Null Spaces of Two Matrices
Let
\[A=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
-1 & -3 & -4
\end{bmatrix} \text{ and }
B=\begin{bmatrix}
1 & 2 & 2 \\
2 &3 &2 \\
5 & 3 & 3
\end{bmatrix}.\]
Determine the null spaces of matrices $A$ and $B$.
Proof.
The null space of the […]

Two Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent
Let $A$ be an $n\times n$ matrix. Suppose that $\lambda_1, \lambda_2$ are distinct eigenvalues of the matrix $A$ and let $\mathbf{v}_1, \mathbf{v}_2$ be eigenvectors corresponding to $\lambda_1, \lambda_2$, respectively.
Show that the vectors $\mathbf{v}_1, \mathbf{v}_2$ are […]

Find All the Eigenvalues of 4 by 4 Matrix
Find all the eigenvalues of the matrix
\[A=\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 &0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{bmatrix}.\]
(The Ohio State University, Linear Algebra Final Exam Problem)
Solution.
We compute the […]

Find a Basis for the Subspace spanned by Five Vectors
Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where
\[
\mathbf{v}_{1}=
\begin{bmatrix}
1 \\ 2 \\ 2 \\ -1
\end{bmatrix}
,\;\mathbf{v}_{2}=
\begin{bmatrix}
1 \\ 3 \\ 1 \\ 1
\end{bmatrix}
,\;\mathbf{v}_{3}=
\begin{bmatrix}
1 \\ 5 \\ -1 […]

Determine When the Given Matrix Invertible
For which choice(s) of the constant $k$ is the following matrix invertible?
\[A=\begin{bmatrix}
1 & 1 & 1 \\
1 &2 &k \\
1 & 4 & k^2
\end{bmatrix}.\]
(Johns Hopkins University, Linear Algebra Exam)
Hint.
An $n\times n$ matrix is […]