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QUESTION

Find the adjoints of the following and, if they exist, the
inverses:

$\left[\begin{array}{ccc}2&amp;-1&amp;4\\-1&amp;0&amp;5\\9&amp;-7&amp;3\end{array}\right],
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\left[\begin{array}{cccc}0&amp;1&amp;1&amp;1\\1&amp;0&amp;1&amp;1\\1&amp;1&amp;0&amp;1\\1&amp;1&amp;1&amp;0\end{array}\right],
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\left[\begin{array}{ccc}1&amp;2&amp;3\\-1&amp;6&amp;-2\\-1&amp;22&amp;0\end{array}\right].$

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ANSWER

First matrix:

adj$A=\left[\begin{array}{ccc}35&amp;-25&amp;-5\\48&amp;-30&amp;-14\\7&amp;5&amp;-1\end{array}\right]$;

$A$adj$A=50I$; $\displaystyle\det A=50;\
A^{-1}=\frac{\textrm{adj}A}{50}$

Second matrix:

adj$A=\left[\begin{array}{cccc}2&amp;-1&amp;-1&amp;-1\\-1&amp;2&amp;-1&amp;-1\\-1&amp;-1&amp;2&amp;-1\\-1&amp;-1&amp;-1&amp;2\end{array}\right]$;

$A$adj$A=-3I$; $\displaystyle\det A=-3;\
A^{-1}=\frac{\textrm{adj}A}{-3}$

Third matrix:

adj$A=\left[\begin{array}{ccc}44&amp;66&amp;-22\\2&amp;3&amp;-1\\-16&amp;-24&amp;8\end{array}\right]$;

$A$adj$A=0$; no inverse.


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