Elementary Matrix and Row Operations

Question

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?

Answer 1

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.

Answer 2

I would look at what the question is asking versus what your calculations.

The question is asking to find a matrix $E$ (the elementary row operation matrix) such that $EA = B$. But in your attempt at the problem you try to find $E$ by solving the equation $AE=B$, which will get you a different solution.

Answer 3

You can simply solve the below equation for $E$.

$$EA=B$$

That is,

$$EA=B \Longrightarrow EAA^{-1}=BA^{-1} \Longrightarrow E=BA^{-1} .$$

So all you need is to find the inverse of $A$, and you're done.