Determine all real numbers $x$ for which there exist $3\times3$ real matrices $A$ and $B$ such that
\begin{align*}
AB -BA = \begin{pmatrix}
1& 0& 0\\
0& x& 0\\
0& 0& -1\\
\end{pmatrix}
\end{align*}
Solution:
Idea: A square matrix $Z$ is called a "commutator" when $Z=XY-YX$ for some matrices $X$ and $Y$. Then the trace($Z$) $= \sum z_{ii}=0$ because the trace($XY$)$=$trace$(YX)$ for all matrices $X$ and $Y$ both whose product $XY$ and $YX$ are square. In this problem the matrix $\begin{pmatrix} 1& 0& 0\\
0& x& 0\\
0& 0& -1\\\end{pmatrix}$ behave as a commutator. So get the trace for both side.
\begin{align*}
\text{tr}(AB -BA) &= \text{tr}\begin{pmatrix}
1& 0& 0\\
0& x& 0\\
0& 0& -1\\
\end{pmatrix}\\
\text{tr}(AB) – \text{tr}(BA) &= x\\
\end{align*}
We know that, there exist any matrices $A$ and $B$ such that tr($AB$) $=$ tr($BA$). So we can say that if $x=0$ then tr$(AB) =$ tr$(BA)$, the above statement holds. If $x\neq 0$ then tr$(AB)$$\neq$ tr$(BA)$, the above statement does not hold
If $x=0$ there exist $3\times3$ real matrices $A$ and $B$ such that tr$(AB) =$ tr$(BA)$
If $x\neq0$ there does not exist $3\times3$ real matrices $A$ and $B$ such that tr$(AB) =$ tr$(BA)$
This is my answer to this problem. But my professor said I did not prove that there are matrices $A, B$ which satisfy the equation in the problem. How do we prove the existence of these matrices?
Best Answer
You can exhibit two matrices $A$ and $B$ that satisfy the given equation when $x=0$. You may try to solve the smaller problem $$ XY-YX=Z:=\pmatrix{1&0\\ 0&-1}\tag{1} $$ first. Then enlarge $X$ and $Y$ to two $3\times3$ matrices $A$ and $B$ by inserting a zero row and a zero column in the middle of each of $X$ and $Y$.
To solve $(1)$, you may pick an $X$ at random (for this particular $Z$ in $(1)$, don't pick a diagonal matrix; do you know why?) and solve for $Y$. Since $Y$ has four entries, you have a system of four linear equations in four unknowns. It's usually solvable unless the choice of $X$ is very bad. One very good choice for our current problem $(1)$ is $X=\pmatrix{0&1\\ 1&0}$.
If you want to know more about the equation $XY-YX=Z$, see Kahan's paper Only Commutators Have Trace Zero.