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
\[XY-YX=I.\]
Add to solve later
Suppose that such matrices exist and consider the trace of the matrix $XY-YX$.
Recall that the trace of a square matrix $A$ is the sum of the diagonal entries of $A$.
Proof.
We use the following basic properties of the trace of matrices.
Let $A, B$ be $n\times n$ matrices and let $c$ be a scalar.
$\tr(A+B)=\tr(A)+\tr(B)$.
$\tr(cA)=c\tr(A)$.
$\tr(AB)=\tr(BA)$.
Seeking a contradiction, we assume that there are matrices $X$ and $Y$ such that $XY-YX=I$.
Then we take the trace of both sides and obtain
\begin{align*}
n&=\tr(I)\\
&=\tr(XY-YX)\\
&=\tr(XY)-\tr(YX) \qquad \text{ (by property (1), (2) of the trace)}\\
&=\tr(XY)-\tr(YX) \qquad \text{ (by property (3) of the trace)}\\
&=0.
\end{align*}
Since $n$ is a positive integer, this is a contradiction.
Therefore, such matrices $X, Y$ do not exist.
The Vector Space Consisting of All Traceless Diagonal Matrices
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 & […]
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 […]
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 […]
Determine Whether Given Matrices are Similar
(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 […]
True or False: If $A, B$ are 2 by 2 Matrices such that $(AB)^2=O$, then $(BA)^2=O$
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.
Proof.
It is true that the matrix $(BA)^2$ must be the zero […]
Trace, Determinant, and Eigenvalue (Harvard University Exam Problem)
(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 […]
Did we assume tr(XY−YX)=n in the proof?
Note that the trace of the $n\times n$ identity matrix is $n$: $\tr(I)=n$.
By assumption, we have $XY-YX=I$. Combining these we have $n=tr(XY-YX)$.