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{\bf Question}

Find the inverses of the following matrices using cofactors:

$$A = \left( \begin{array}{cccc}
 1 &amp; 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 1 &amp; 0 \\ 1 &amp; 1 &amp; 0 &amp; 0 \\ 1 &amp; 0 &amp; 0 &amp; 0 \end{array}
 \right) \hspace{.2in} B = \left( \begin{array}{ccc} 2 &amp; 1 &amp; 3 \\ 1 &amp; 1 &amp; 2 \\ 2 &amp; 1 &amp; 6
\end{array} \right)$$

\vspace{.25in}

{\bf Answer}

Inverse of $A = \left( \begin{array}{cccc} 1 &amp; 0 &amp; 0 &amp; 1 \\ 1 &amp; 0
&amp; 1 &amp; 0\\1 &amp; 1 &amp; 0 &amp; 0\\1 &amp; 0 &amp; 0 &amp; 0 \end{array} \right)$

det(a) = 1

$A^{-1} = \frac {adj(A)}{|A|}$


Matrix of cofactors
\begin{eqnarray*} [A_{ij}] &amp; = &amp; \left[ \begin{array}{cccc}
\left|\begin{array}{ccc} 0 &amp; 1 &amp; 0\\ 1 &amp; 0 &amp;0 \\ 0 &amp; 0 &amp; 0
\end{array}\right| &amp;
-\left|\begin{array}{ccc} 1 &amp; 1 &amp; 0\\ 1 &amp; 0 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
\left|\begin{array}{ccc} 1 &amp; 0 &amp; 0\\ 1 &amp; 1 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
-\left|\begin{array}{ccc} 1 &amp; 0 &amp; 1\\ 1 &amp; 1 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| \\ \\
-\left|\begin{array}{ccc} 0 &amp; 0 &amp; 1\\ 1 &amp; 0 &amp;0 \\ 0 &amp; 0 &amp; 0
\end{array}\right| &amp;
\left|\begin{array}{ccc} 1 &amp; 0 &amp; 1\\ 1 &amp; 0 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
-\left|\begin{array}{ccc} 1 &amp; 0 &amp; 1\\ 1 &amp; 1 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
\left|\begin{array}{ccc} 1 &amp; 0 &amp; 0\\ 1 &amp; 1 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| \\ \\
\left|\begin{array}{ccc} 0 &amp; 0 &amp; 1\\ 0 &amp; 1 &amp;0 \\ 0 &amp; 0 &amp; 0
\end{array}\right| &amp;
-\left|\begin{array}{ccc} 1 &amp; 0 &amp; 1\\ 1 &amp; 1 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
\left|\begin{array}{ccc} 1 &amp; 0 &amp; 1\\ 1 &amp; 0 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
-\left|\begin{array}{ccc} 1 &amp; 0 &amp; 0\\ 1 &amp; 0 &amp;1 \\ 1 &amp; 0 &amp; 0
\end{array}\right| \\ \\
-\left|\begin{array}{ccc} 0 &amp; 0 &amp; 1\\ 0 &amp; 1 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
\left|\begin{array}{ccc} 1 &amp; 0 &amp; 1\\ 1 &amp; 1 &amp;0 \\ 1 &amp; 0 &amp; 0
\end{array}\right| &amp;
-\left|\begin{array}{ccc} 1 &amp; 0 &amp; 1\\ 1 &amp; 0 &amp;0 \\ 1 &amp; 1 &amp; 0
\end{array}\right| &amp;
\left|\begin{array}{ccc} 1 &amp; 0 &amp; 0\\ 1 &amp; 0 &amp;1 \\ 1 &amp; 1 &amp; 0
\end{array}\right| \\ \\
\end{array}\right]\\ &amp; = &amp; \left( \begin{array}{cccc} 0 &amp; 0 &amp; 0 &amp; 1 \\
 0 &amp; 0 &amp; 1 &amp; 0 \\ 0 &amp; 1 &amp; 0 &amp; 0 \\ 1 &amp; -1 &amp; -1 &amp; -1 \end{array} \right) \end{eqnarray*}

Hence $A^{-1} = \left( \begin{array}{cccc} 0 &amp; 0 &amp; 0 &amp; 1 \\
 0 &amp; 0 &amp; 1 &amp; -1 \\ 0 &amp; 1 &amp; 0 &amp; -1 \\ 1 &amp; 0 &amp; 0 &amp; -1 \end{array}
 \right)$

Finding the inverse of $B = \left( \begin{array}{ccc} 2 &amp; 1 &amp; 3
\\ 1 &amp; 1 &amp; 2 \\ 2 &amp; 1 &amp; 6 \end{array} \right)$

Matrix of cofactors
\begin{eqnarray*} [A_{ij}] &amp; = &amp; \left[ \begin{array}{ccc}
\left|\begin{array}{cc} 1 &amp; 2 \\ 1 &amp; 6 \end{array}\right| &amp;
-\left|\begin{array}{cc} 1 &amp; 2 \\ 2 &amp; 6 \end{array}\right| &amp;
\left|\begin{array}{cc} 1 &amp; 1 \\ 2 &amp; 1 \end{array}\right| \\ \\
-\left|\begin{array}{cc} 1 &amp; 3 \\ 1 &amp; 6 \end{array}\right| &amp;
\left|\begin{array}{cc} 2 &amp; 3 \\ 2 &amp; 6 \end{array}\right| &amp;
-\left|\begin{array}{cc} 2 &amp; 1 \\ 2 &amp; 1 \end{array}\right| \\ \\
\left|\begin{array}{cc} 1 &amp; 3 \\ 1 &amp; 2 \end{array}\right| &amp;
-\left|\begin{array}{cc} 2 &amp; 3 \\ 1 &amp; 2 \end{array}\right| &amp;
\left|\begin{array}{cc} 2 &amp; 1 \\ 1 &amp; 1 \end{array}\right| \\
\end{array}\right]\\ &amp; = &amp; \left( \begin{array}{ccc} 4 &amp; -2 &amp; -1 \\
 -3 &amp; 6 &amp; 0 \\ -1 &amp; -1 &amp; 1\end{array} \right) \end{eqnarray*}

Hence $B^{-1} = \frac{1}{3} \left( \begin{array}{ccc} 4 &amp; -3 &amp; -1
\\ -2 &amp; 6 &amp; -1 \\ -1 &amp; 0 &amp; 1 \end{array} \right)$



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