Matrix HELP! - Finding Matrices for AM = -MA

[theacerf1]

Matrix HELP!

Ok this question is in a cd quiz from a textbook, i have no idea what the question even means or how to do it. Please help!

Let M =

 7 −3
15 −7 .

Find all 2 × 2 matrices

A =
[ a b
b d ]

such that AM = −MA.
(Note that the solutions wanted should have equal (1, 2)- and (2, 1)-entries b.)

 

[theacerf1]

M =
[7 -3
15 -7]

a 2x2 matrix

 

[HallsofIvy]

Did you even try? It is really just a matter of computation.

$$\begin{bmatrix}a & b \\ b & d\end{bmatrix}\begin{bmatrix}7 & -3 \\ 15 & -7\end{bmatrix}$$
$$= \begin{bmatrix}7a+ 15b & -3a- 7b \\ 7b+ 15d & -3b- 7d\end{bmatrix}$$
$$= -\begin{bmatrix}7 & -3 \\ 15 & -7\end{bmatrix}\begin{bmatrix}a & b \\ b & d\end{bmatrix}$$
$$= \begin{bmatrix}-7a+ 3b & -7b+ 3d \\ -15a+ 7b & -15b+ 7d\end{bmatrix}$$

So you must have 7a+ 15b= -7a+ 3b, -3a- 7b= -7b+ 3d, 7b+ 15d= -15a+ 7d, and -3b- 7d= -15b+ 7d.