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So I have this matrix

A = $\begin{bmatrix}2 & 4\\1 & 1\end{bmatrix}$

I am tasked with finding all the elementary matrices such that Ek...E2E1A = I. Use this sequence to write both A and A-1 as products of elementary matrices/

I ended up getting four elementary matrices

E1 = $\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$ E2 = $\begin{bmatrix}1 & 0\\-2 & 1\end{bmatrix}$ E3 = $\begin{bmatrix}1 & 0\\0 & 1/2\end{bmatrix}$ E4 = $\begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$

When I invert all the elementary matrices and multiply them it equals A but when I multiple All the elementary matrices by A I do not get the identity matrix

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HINT Possibly you are multiplying in the wrong order. Remember in general $$(AB)^{-1} = B^{-1} A^{-1} \neq A^{-1} B^{-1}$$

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A matrix is an elementary if it differs from the identity matrix by a single elementary row or column operation.

Elementary Matrix


The reduction process is $$ % \left[ \begin{array}{c|c} \mathbf{A} & \mathbf{I} \end{array} \right] % \qquad \Rightarrow \qquad % \left[ \begin{array}{c|c} \mathbf{E_{A}} & \mathbf{R} \end{array} \right] $$

First form the augmented matrix $$ \left[ \begin{array}{c|c} \mathbf{A} & \mathbf{I} \end{array} \right] = \left[ \begin{array}{cc|cc} 2 & 4 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ \end{array} \right] $$

Normalize row 1: $$ \left[ \begin{array}{cc} \frac{1}{2} & 0 \\ 0 & 1 \\ \end{array} \right] % \left[ \begin{array}{cc|cc} 2 & 4 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ \end{array} \right] = \left[ \begin{array}{cc|cc} 1 & 2 & \frac{1}{2} & 0 \\ 1 & 1 & 0 & 1 \\ \end{array} \right] $$

Clear column 1 $$ \left[ \begin{array}{rc} 1 & 0 \\ -1 & 1 \\ \end{array} \right] % \left[ \begin{array}{cc|cc} 1 & 2 & \frac{1}{2} & 0 \\ 1 & 1 & 0 & 1 \\ \end{array} \right] = \left[ \begin{array}{cr|rc} 1 & 2 & \frac{1}{2} & 0 \\ 0 & -1 & -\frac{1}{2} & 1 \\ \end{array} \right] $$

Normalize row 2 $$ \left[ \begin{array}{cr} 1 & 0 \\ 0 & -1 \\ \end{array} \right] % \left[ \begin{array}{cc|cr} 1 & 2 & \frac{1}{2} & 0 \\ 0 & 1 & \frac{1}{2} & -1 \\ \end{array} \right] = \left[ \begin{array}{cc|cr} 1 & 2 & \frac{1}{2} & 0 \\ 0 & 1 & \frac{1}{2} & -1 \\ \end{array} \right] $$

Clear column 2 $$ \left[ \begin{array}{cr} 1 & -2 \\ 0 & 1 \\ \end{array} \right] % \left[ \begin{array}{cc|cr} 1 & 2 & \frac{1}{2} & 0 \\ 0 & 1 & \frac{1}{2} & -1 \\ \end{array} \right] = \left[ \begin{array}{cc|rr} 1 & 0 & -\frac{1}{2} & 2 \\ 0 & 1 & \frac{1}{2} & -1 \\ \end{array} \right] $$


Product of the elementary matrices $$ % \begin{align} % \mathbf{E}_{4} \, \mathbf{E}_{3} \, \mathbf{E}_{2} \, \mathbf{E}_{1} \mathbf{A} &= \mathbf{I}_{2} \\[3pt] % % four \left[ \begin{array}{cr} 1 & -2 \\ 0 & 1 \\ \end{array} \right] % third \left[ \begin{array}{cr} 1 & 0 \\ 0 & -1 \\ \end{array} \right] % second \left[ \begin{array}{rc} 1 & 0 \\ -1 & 1 \\ \end{array} \right] % first \left[ \begin{array}{cc} \frac{1}{2} & 0 \\ 0 & 1 \\ \end{array} \right] % A \left[ \begin{array}{cc|cc} 2 & 4 \\ 1 & 1 \\ \end{array} \right] &= \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \\ \end{array} \right] % \end{align} % $$

$$ % \begin{align} % \mathbf{E}_{4} \, \mathbf{E}_{3} \, \mathbf{E}_{2} \, \mathbf{E}_{1} \mathbf{I}_{2} &= \mathbf{A}^{-1} \\[3pt] % % four \left[ \begin{array}{cr} 1 & -2 \\ 0 & 1 \\ \end{array} \right] % third \left[ \begin{array}{cr} 1 & 0 \\ 0 & -1 \\ \end{array} \right] % second \left[ \begin{array}{rc} 1 & 0 \\ -1 & 1 \\ \end{array} \right] % first \left[ \begin{array}{cc} \frac{1}{2} & 0 \\ 0 & 1 \\ \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \\ \end{array} \right] &= \left[ \begin{array}{rr} -\frac{1}{2} & 2 \\ \frac{1}{2} & -1 \\ \end{array} \right] % \end{align} % $$

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