Science:Math Exam Resources/Courses/MATH152/April 2022/Question A19

MATH152 April 2022

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Question A19
Find the inverse of the matrix $$A=\left[{\begin{array}{cccc}-1&0&0&3\\0&2&0&0\\0&0&{\frac {1}{2}}&0\\0&0&0&1\end{array}}\right].$$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The inverse $A^{-1}$ is a matrix that satisfies the equation $AA^{-1}=\mathrm {Id}$. Recall that you can solve this (system of) equation(s) by using row operations on an augmented matrix. See section 4.5 of the notes for MATH 152, |linked here.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Recall, from the hint, that row operations can be used to write the inverse of a matrix. We set up the augmented matrix ${\textstyle [A\ |\ I]}$, once transformed using row operations so that the identity is on the left, it necessarily shows ${\textstyle A^{-1}}$ on the right, ${\textstyle [I\ |\ A^{-1}]}$. $$\left[{\begin{array}{cccc|cccc}-1&0&0&3&1&0&0&0\\0&2&0&0&0&1&0&0\\0&0&{\frac {1}{2}}&0&0&0&1&0\\0&0&0&1&0&0&0&1\end{array}}\right]\rightarrow \left[{\begin{array}{cccc|cccc}-1&0&0&0&1&0&0&-3\\0&2&0&0&0&1&0&0\\0&0&{\frac {1}{2}}&0&0&0&1&0\\0&0&0&1&0&0&0&1\end{array}}\right]\rightarrow \left[{\begin{array}{cccc|cccc}1&0&0&0&-1&0&0&3\\0&1&0&0&0&{\frac {1}{2}}&0&0\\0&0&1&0&0&0&2&0\\0&0&0&1&0&0&0&1\end{array}}\right].$$ We conclude $$A^{-1}=\left[{\begin{array}{cccc}-1&0&0&3\\0&{\frac {1}{2}}&0&0\\0&0&2&0\\0&0&0&1\end{array}}\right].$$