Matrix multiplication and determinant question Show that if $\det(\begin{bmatrix}b & c\\a & b\end{bmatrix})=0$ with  $A=\begin{bmatrix}a & a\\b & b\end{bmatrix}$ and $B=\begin{bmatrix}b & b\\c & c\end{bmatrix}$ then $AB=BA.$
How do I go by solving this? I tried finding the determent, $AB$, and $BA$ but it didn't work.

$$
\begin{bmatrix} a & a \\ b & b \end{bmatrix} \begin{bmatrix} b & b \\ c & c \end{bmatrix} = \begin{bmatrix} ab+ac & ab+ac \\ c^2+bc & b^2 + bc \end{bmatrix} 
$$
$$
\begin{bmatrix} b & b \\ c & c \end{bmatrix} \begin{bmatrix} a & a \\ b & b \end{bmatrix} = \begin{bmatrix} ab+b^2 & ab+b^2 \\ ac+bc & ac + bc \end{bmatrix} 
$$
So we ask whether $ab+ac$ is the same as $ab+b^2$.  They're the same if $ac=b^2$.  That happens if $b^2-ac=0$.  And $b^2-ac$ is the determinant that you mentioned.  So check the other three entries in the same way.
