Recall, from the hint, that row operations can be used to write the inverse of a matrix. We set up the augmented matrix [ A | I ] {\textstyle [A\ |\ I]} , once transformed using row operations so that the identity is on the left, it necessarily shows A − 1 {\textstyle A^{-1}} on the right, [ I | A − 1 ] {\textstyle [I\ |\ A^{-1}]} .
We conclude
![{\textstyle [A\ |\ I]}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/8c8f582dd5792edf6440a7411d3b28d97abfb73f)
, once transformed using row operations so that the identity is on the left, it necessarily shows 
on the right, ![{\textstyle [I\ |\ A^{-1}]}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/cf78c8b4f933196bd5a9ac18a40f88696aaef64c)
.
![{\displaystyle \left[{\begin{array}{cccc|cccc}-1&0&0&3&1&0&0&0\\0&2&0&0&0&1&0&0\\0&0&{\frac {1}{2}}&0&0&0&1&0\\0&0&0&1&0&0&0&1\end{array}}\right]\rightarrow \left[{\begin{array}{cccc|cccc}-1&0&0&0&1&0&0&-3\\0&2&0&0&0&1&0&0\\0&0&{\frac {1}{2}}&0&0&0&1&0\\0&0&0&1&0&0&0&1\end{array}}\right]\rightarrow \left[{\begin{array}{cccc|cccc}1&0&0&0&-1&0&0&3\\0&1&0&0&0&{\frac {1}{2}}&0&0\\0&0&1&0&0&0&2&0\\0&0&0&1&0&0&0&1\end{array}}\right].}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/9e53f01cbc5b0c409034ae1cdf028c4937414dd3)
![{\displaystyle A^{-1}=\left[{\begin{array}{cccc}-1&0&0&3\\0&{\frac {1}{2}}&0&0\\0&0&2&0\\0&0&0&1\end{array}}\right].}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/9bc42e488e0faaa731ba58d81c5aeab85c56209e)
