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QUESTION

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

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

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ANSWER

First matrix:

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

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

Second matrix:

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

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

Third matrix:

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

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


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