Given that A = $\begin{bmatrix} 2 & 1 \\ -5 & -4 \end{bmatrix} $ and B = $\begin{bmatrix} 3 & -1 \\ -1 & 0 \end{bmatrix} $
Find a 2 X 2 matrix C such that $CA= B$
I multiply both sides by $A^{-1}$
Since $A^{-1}A = I $
$ CI = BA^{-1}$
Since $CI = IC = C$
$ C = BA^{-1} $
However, when I carry on and find out the answer to matrix C, I can’t get the answer. Where have I gone wrong ?
The inverse matrix of $A$ is $$A^{-1}=\left[\begin{matrix}\frac 43&\frac 13\\ -\frac {5}3&-\frac {2}3\end{matrix}\right]$$ And then the matrix $C$ is $$C=BA^{-1}=\left[\begin{matrix}3&-1\\ -1&0\end{matrix}\right]\left[\begin{matrix}\frac 43&\frac 13\\ -\frac {5}3&-\frac {2}3\end{matrix}\right]=\left[\begin{matrix}\frac {17}3&\frac 32\\ -\frac {4}3&-\frac {1}3\end{matrix}\right]$$
Matrices aren’t a commutative group. Read this.