# Give the elementary matrix that converts [-2,-2,-1,-3,-1,-3,1,-4,-3] to [-6,-2,-1,-5,-1,-3,-7,-4,-3]

Matrix transformations

Give the elementary matrix that converts
$$\begin{bmatrix} -2 & -2 & -1 \\ -3 & -1 & -3 \\ 1 & -4 & -3 \end{bmatrix}$$

to

$$\begin{bmatrix} -6 & -2 & -1 \\ -5 & -1 & -3 \\ -7 & -4 & -3 \end{bmatrix}$$

2021-03-08

Notice that the difference between the two matrices is that the latter's first row is
$$\left(\begin{array}{c} -2\\ -3 \\ 1\end{array}\right)+2\left(\begin{array}{c} -2 \\ -1 \\ -4 \end{array}\right)=\left(\begin{array}{c} -6 \\ -5 \\ -7 \end{array}\right)$$
Looking over your elementary matrix types, we see that the one that adds 2 times the second row to the first looks something like this.
$$\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
That is, to add row k times row ii to row j, you take the identity matrix and add a k to column ii and row j.
We can multiply it out to confirm
$$\begin{bmatrix} -6 & -2 & -1 \\ -5 & -1 & -3 \\ -7 & -4 & -3 \end{bmatrix}$$