# Let A=begin{pmatrix}2 &1 6 & 4 end{pmatrix} a) Express A^{-1} as a product of elementary matrices b) Express A as a product of elementary matrices

Question
Matrices
Let $$A=\begin{pmatrix}2 &1 \\6 & 4 \end{pmatrix}$$
a) Express $$A^{-1}$$ as a product of elementary matrices
b) Express A as a product of elementary matrices

2020-11-18
Step 1
Given: $$A=\begin{pmatrix}2 &1 \\6 & 4 \end{pmatrix}$$
a) Express $$A^{-1}$$ as a product of elementary matrices
b) Express A as a product of elementary matrices
Step 2
$$\begin{bmatrix}2 & 1 \\6 & 4 \end{bmatrix} \begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} R_2 : R_2-3R_1 \begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} R_2 : R_2-3R_1=\begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix}E$$
$$=\begin{bmatrix}2 & 1 \\0 & 1 \end{bmatrix} \begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix} R_2 : R_2 \div 2 = \begin{bmatrix}1/2 & 0 \\0 & 1 \end{bmatrix}E_2$$
$$=\begin{bmatrix}1 & 1/2 \\0 & 1 \end{bmatrix} \begin{bmatrix}1/2 & 0 \\-3 & 1 \end{bmatrix}R_1=R_1-\frac{R_2}{2}\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}R_1:R_1-\frac{R_2}{2}=\begin{bmatrix}1 & -1/2 \\ 0 & 1 \end{bmatrix}E_3$$
$$=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} \begin{bmatrix}2 & -1/2 \\-3 & 1 \end{bmatrix}$$
$$\therefore E_3 E_2 E_1 A =I$$
$$E_3 E_2 E_1I =A^{-1}$$
you can recheck it
$$A^{-1}=E_3E_2E_1=\begin{bmatrix}1 & -1/2 \\0 & 1 \end{bmatrix}\begin{bmatrix}1/2 & 0 \\0 & 1 \end{bmatrix}\begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}1 & -1/2 \\0 & 1 \end{bmatrix}\begin{bmatrix} 1/2 & 0 \\-3 & 1 \end{bmatrix} =\begin{bmatrix} 1/2 & 3/2 \\-3 & 1 \end{bmatrix}$$
$$=\begin{bmatrix} 2 & -1/2 \\-3 & 1 \end{bmatrix}$$
Step 3
$$(A^{-1})^{-1}=(E_3E_2E_1)^{-1}$$
$$A=E_1^{-1}E_2^{-1}E_3^{-1}$$
$$E_1=\begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix} , E_1^{-1}=\frac{1}{1}\begin{bmatrix}1 & 0 \\3 & 1 \end{bmatrix}$$
$$E_2=\begin{bmatrix}1/2 & 0 \\0 & 1 \end{bmatrix} , E_2^{-1}=\frac{1}{1/2}=\begin{bmatrix}1 & 0 \\0 & 1/2 \end{bmatrix}$$
$$E_3=\begin{bmatrix}1 & -1/2 \\0 & 1 \end{bmatrix} , E_3^{-1}=\frac{1}{1}=\begin{bmatrix}1 & 1/2 \\0 & 1 \end{bmatrix}$$
$$A=2\begin{bmatrix}1 & 0 \\3 & 1 \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1/2 \end{bmatrix}\begin{bmatrix}1 & 1/2 \\0 & 1 \end{bmatrix}$$
$$=2\begin{bmatrix}1 & 0 \\3 & 1 \end{bmatrix}\begin{bmatrix}1 & 1/2 \\0 & 1/2 \end{bmatrix}=2\begin{bmatrix}1 & 1/2 \\3 & 3/2+1/2 \end{bmatrix}$$
$$=\begin{bmatrix}2 & 1 \\6 & 4 \end{bmatrix}$$

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