Find an elementary matrix E such that EA = B
$$A = \begin{bmatrix} 2 & -1 \\ 5 & 3 \end{bmatrix}$$
$$B = \begin{bmatrix} -4 & 2 \\ 5 & 3 \end{bmatrix}$$
$$\begin{bmatrix} 2 & -1 \\ 5 & 3 \end{bmatrix}*E=\begin{bmatrix} -4 & 2 \\ 5 & 3 \end{bmatrix}$$
So the Elementary Matrix is an Identity Matrix that has one elementary row operation performed on it. Multiplying an Elementary Matrix by A should result in a matrix that is equivalent to having that elementary row operation performed onto A.
I can see that only the first row of A is modified to obtain B and I can tell that the first row of A is scaled by a value of -2.
Therefore the Elementary Matrix should be the Identity Matrix with the first row scaled by -2.$$\begin{bmatrix} -2 & 0 \\ 0 & 1 \end{bmatrix}$$
However, $$\begin{bmatrix} 2 & -1 \\ 5 & 3 \end{bmatrix}*\begin{bmatrix} -2 & 0 \\ 0 & 1 \end{bmatrix}≠\begin{bmatrix} -4 & 2 \\ 5 & 3 \end{bmatrix}$$
What am I doing incorrectly?
Best Answer
Since the operation is $B=EA$ we have \begin{equation} \begin{pmatrix} -2 & 0\\0 & 1 \end{pmatrix}\begin{pmatrix} 2 & -1\\5 & 3 \end{pmatrix} = \begin{pmatrix} -4 & 2\\5 & 3 \end{pmatrix}. \end{equation} Your elementary matrix is correct but you meant to multiply it to $A$ on the left not on the right.