Question

Find Elementary matrices E1, E2 so that E2 E1 A = I2, where A = matrix(1 0 2 3) and I2 is the respective identity matrix

Answer 1

$Given \space that\\A = \begin{bmatrix}1 & 0 & \\2 & 3\end{bmatrix}$

Find elementary Matrices E1 and E2 such that E2 E1 A = I

Start by eliminating the 2 in matrix A.

$E_1A = \begin{bmatrix}1 & 0 & \\0 & 3\end{bmatrix}\\\space\\ \begin{bmatrix}a & b & \\c & d\end{bmatrix}\begin{bmatrix}1 & 0 & \\2 & 3\end{bmatrix} = \begin{bmatrix}1 & 0 & \\0 & 3\end{bmatrix}\\\space\\ \begin{bmatrix}a+2b & 3b & \\c+2d & 3d\end{bmatrix} =\begin{bmatrix}1 & 0 & \\0 & 3\end{bmatrix}$

hence a=1; b= 0; c=-2 ; d= 1

Therefore E1 = $\begin{bmatrix}1 & 0 & \\-2 & 1\end{bmatrix}$ [Answer]

Find E2 (since we know what E1A is)

$\begin{bmatrix}a & b & \\c & d\end{bmatrix}\begin{bmatrix}1 & 0 & \\0 & 3\end{bmatrix} = \begin{bmatrix}1 & 0 & \\0 & 1\end{bmatrix}\\\space\\ \begin{bmatrix}a & 3b & \\c & 3d\end{bmatrix} = \begin{bmatrix}1 & 0 & \\0 & 1\end{bmatrix}\\\space\\$

hence a= 1; b= 0; c= 0; d=$\frac{1}{3}$

Therefore E2 = $\begin{bmatrix}1 & 0 & \\0 & \frac{1}{3}\end{bmatrix}$ [Answer]