# 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

Let $A=\left(\begin{array}{cc}2& 1\\ 6& 4\end{array}\right)$
a) Express ${A}^{-1}$ as a product of elementary matrices
b) Express A as a product of elementary matrices
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Step 1
Given: $A=\left(\begin{array}{cc}2& 1\\ 6& 4\end{array}\right)$
a) Express ${A}^{-1}$ as a product of elementary matrices
b) Express A as a product of elementary matrices
Step 2
$\left[\begin{array}{cc}2& 1\\ 6& 4\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]{R}_{2}:{R}_{2}-3{R}_{1}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]{R}_{2}:{R}_{2}-3{R}_{1}=\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right]E$
$=\left[\begin{array}{cc}2& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right]{R}_{2}:{R}_{2}÷2=\left[\begin{array}{cc}1/2& 0\\ 0& 1\end{array}\right]{E}_{2}$
$=\left[\begin{array}{cc}1& 1/2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1/2& 0\\ -3& 1\end{array}\right]{R}_{1}={R}_{1}-\frac{{R}_{2}}{2}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]{R}_{1}:{R}_{1}-\frac{{R}_{2}}{2}=\left[\begin{array}{cc}1& -1/2\\ 0& 1\end{array}\right]{E}_{3}$
$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}2& -1/2\\ -3& 1\end{array}\right]$
$\therefore {E}_{3}{E}_{2}{E}_{1}A=I$
${E}_{3}{E}_{2}{E}_{1}I={A}^{-1}$
you can recheck it
${A}^{-1}={E}_{3}{E}_{2}{E}_{1}=\left[\begin{array}{cc}1& -1/2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1/2& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right]$
$=\left[\begin{array}{cc}1& -1/2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1/2& 0\\ -3& 1\end{array}\right]=\left[\begin{array}{cc}1/2& 3/2\\ -3& 1\end{array}\right]$
$=\left[\begin{array}{cc}2& -1/2\\ -3& 1\end{array}\right]$
Step 3
$\left({A}^{-1}{\right)}^{-1}=\left({E}_{3}{E}_{2}{E}_{1}{\right)}^{-1}$
$A={E}_{1}^{-1}{E}_{2}^{-1}{E}_{3}^{-1}$
${E}_{1}=\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right],{E}_{1}^{-1}=\frac{1}{1}\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right]$
${E}_{2}=\left[\begin{array}{cc}1/2& 0\\ 0& 1\end{array}\right],{E}_{2}^{-1}=\frac{1}{1/2}=\left[\begin{array}{cc}1& 0\\ 0& 1/2\end{array}\right]$
${E}_{3}=\left[\begin{array}{cc}1& -1/2\\ 0& 1\end{array}\right],{E}_{3}^{-1}=\frac{1}{1}=\left[\begin{array}{cc}1& 1/2\\ 0& 1\end{array}\right]$
Jeffrey Jordon