Express the Eigenvalues of a 2 by 2 Matrix in Terms of the Trace and Determinant
Let $A=\begin{bmatrix}
a & b\\
c& d
\end{bmatrix}$ be an $2\times 2$ matrix.
Express the eigenvalues of $A$ in terms of the trace and the determinant of $A$.
Solution.
Recall the definitions of the trace and determinant of $A$:
\[\tr(A)=a+d \text{ and } […]

Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?
Let $A$ be an $n \times n$ matrix.
Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.
Solution.
The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries. Also, note that the […]

If Two Matrices are Similar, then their Determinants are the Same
Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.
Proof.
Suppose that $A$ and $B$ are similar. Then there exists a nonsingular matrix $S$ such that
\[S^{-1}AS=B\]
by definition.
Then we […]

An Example of a Matrix that Cannot Be a Commutator
Let $I$ be the $2\times 2$ identity matrix.
Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.
Proof.
Assume that $[A, B]=-I$. Then $ABA^{-1}B^{-1}=-I$ implies
\[ABA^{-1}=-B. […]

If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix
Let $A, B$ be complex $2\times 2$ matrices satisfying the relation
\[A=AB-BA.\]
Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.
Hint.
Find the trace of $A$.
Use the Cayley-Hamilton theorem
Proof.
We first calculate the […]

Matrix $XY-YX$ Never Be the Identity Matrix
Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that
\[XY-YX=I.\]
Hint.
Suppose that such matrices exist and consider the trace of the matrix $XY-YX$.
Recall that the trace of […]

Matrices Satisfying $HF-FH=-2F$
Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation
\[HF-FH=-2F.\]
(a) Find the trace of the matrix $F$.
(b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ […]