Let $A$ and $B$ be $n \times n$ matrices.
Is it always true that $\tr (A B) = \tr (A) \tr (B) $?
If it is true, prove it. If not, give a counterexample.
Add to solve later 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.
Add to solve later Let $A$ and $B$ be $2\times 2$ matrices such that $(AB)^2=O$, where $O$ is the $2\times 2$ zero matrix.
Determine whether $(BA)^2$ must be $O$ as well. If so, prove it. If not, give a counter example.
Add to solve later (a) Is the matrix $A=\begin{bmatrix}
1 & 2\\
0& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
1& 2
\end{bmatrix}$?
(b) Is the matrix $A=\begin{bmatrix}
0 & 1\\
5& 3
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}$?
(c) Is the matrix $A=\begin{bmatrix}
-1 & 6\\
-2& 6
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
3 & 0\\
0& 2
\end{bmatrix}$?
(d) Is the matrix $A=\begin{bmatrix}
-1 & 6\\
-2& 6
\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
1 & 2\\
-1& 4
\end{bmatrix}$?
Add to solve later Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.
Add to solve later (a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.
Find $\det(A)$.
(b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
(c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of $A$?
(Harvard University, Linear Algebra Exam Problem)
Add to solve later 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.
Add to solve later 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
\[XY-YX=I.\]
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Add to solve later Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.
That is,
\begin{equation*}
V:=\left\{ A=\begin{bmatrix}
a_{11} & 0 & \dots & 0 \\
0 &a_{22} & \dots & 0 \\
0 & 0 & \ddots & \vdots \\
0 & 0 & \dots & a_{nn}
\end{bmatrix} \quad \middle| \quad
\begin{array}{l}
a_{11}, \dots, a_{nn} \in \C,\\
\tr(A)=0 \\
\end{array}
\right\}
\end{equation*}
Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.
(a) Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)
(b) Show that matrices
\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.
(c) Find the dimension of $V$.
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Add to solve later Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that
(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$
(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$
Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$.
Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues.
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