1. ## Matrix Problem

Hey guys, I've been having trouble with this matrix problem for a while now.

Here it is:

M = $\left[ \begin{array}{cc} 7 & -3 \\ 15 & -7 \end{array} \right]$

Find all 2 x 2 matrices A = $\left[ \begin{array}{cc} a & b \\ b & d \end{array} \right]$ where (1,2) = (2,1) entry such that AM = -MA

I've tried matrix multiplication for both AM and -MA and tried equating appropriate values but I end up with a zero matrix. I also tried multiplying both sides by A inverse, but again I get a zero matrix. Any help is appreciated, thanks.

2. \displaystyle \begin{align*}\mathbf{AM} &= \left[\begin{matrix} a & b\\ b & d\end{matrix}\right]\left[\begin{matrix} \phantom{1}7 & -3 \\ 15 & -7\end{matrix}\right]\\ &= \left[\begin{matrix}7a + 15b & -3a - 7b\\ 7b + 15d & -3b - 7d\end{matrix}\right]\\ \\ \\ -\mathbf{MA} &= -\left[\begin{matrix}\phantom{1}7 & -3\\ 15 & -7\end{matrix}\right]\left[\begin{matrix}a & b\\ b & d\end{matrix}\right]\\ &= -\left[\begin{matrix}7a - 3b & 7b - 3d \\ 15a - 7b & 15b - 7d\end{matrix}\right] \\ &= \left[\begin{matrix}3b- 7a & 3d - 7b \\ 7b - 15a & 7d - 15b\end{matrix}\right]\end{align*}

When you equate these matrices, by equating the corresponding elements you should be able to solve for a, b, d.