avissidep

2021-01-17

Find the inverse of the matrix using elementary matrices. $\left[\left[2,0\right],\left[1,1\right]\right]$

Cristiano Sears

Let the given matrix be,
$A=\left[\left[2,0\right],\left[1,1\right]\right]$
Now transform the given matrix in to unit matrix I by performing column reduction to get,
$A=\left[\left[2,0\right],\left[1,1\right]\right]$
$=\left[\left[2,0\right],\left[1,1\right]\right]\left({c}_{1}->{c}_{1}-{c}_{2}\right)$
$=\left[\left[1,0\right],\left[0,1\right]\right]\left({c}_{1}->1/2\ast {c}_{1}\right)$
=I
Now, construct sequence of elementary matrices such that E2*E1 A=I
The column operation are,
${E}_{1}=\left[\left[1,0\right],\left[0,1\right]\right]\left({c}_{1}->{c}_{1}-{c}_{2}\right)$
$=\left[\left[1,0\right],\left[-1,1\right]\right]$
${E}_{2}=\left[\left[1,0\right],\left[0,1\right]\right]\left({c}_{1}->1/2\ast {c}_{1}\right)$
$=\left[\left[1/2,0\right],\left[0,1\right]\right]$
Then the inverse of the matrix by elementary matrices is given by:
${A}^{-1}={E}_{1}×{E}_{2}$
$=\left[\left[1,0\right],\left[-1,1\right]\right]\ast \left[\left[1/2,0\right],\left[0,1\right]\right]$
$=\left[\left[1/2,0\right],\left[-1/2,1\right]\right]$

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