Background

I saw a question about determinants.

Suppose that $A$ and $B$ are singular and nonsingular matrices respectively. Simplify $\det((A+B)^2−(A−B)^2)$.

Problem

A wrong solution with a vote of -2 is chosen by Daniel. Why can this happen?

Possible explanation

That’s because he’s correctly done the expansion until $\det(2AB + 2BA)$.

Raison d’être of this post

Having spent time on typing a comment, I worry that it will automatically disappear in sooner or later if the accepted answer is deleted. Therefore, I back it up here.

Consider $A = \begin{bmatrix} 1&2&3\\ 0&0&0\\ 5&7&9 \end{bmatrix} , B = \begin{bmatrix} 3&2&3\\ 2&2&1\\ 3&1&3 \end{bmatrix}.$ $\begin{align*} \det(AB+BA) &= \det\left( \begin{bmatrix} 16&9&14 \\ 0&0&0 \\ 56&33&49 \end{bmatrix} +\begin{bmatrix} 18&27&36 \\ 7&11&15 \\ 18&27&36 \end{bmatrix} \right) \\ &=\det\left( \begin{bmatrix} 34&36&50\\ 7&11&15\\ 74&60&85 \end{bmatrix} \right)= 30 \ne 0. \end{align*}$ However, if $A = 0$ and $B = I_3$, then the answer is clearly zero. As a result, we can’t decude further from $\det(2(AB + BA))$.

Lessons learnt

The generation of a random matrix/array of integers using randi([imin, imax], m, n). For more details, you may read GNU Octave’s manual.