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{\bf Question}

Write down the matrix of the minors $M_{ij}$ and the matrix of
cofactors $A_{ij} $ associated with the matrix $A= \left(
\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 1 & 1
\end{array} \right)$ with elements $a_{ij}$

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{\bf Answer}

$A= \left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 1 & 1
\end{array} \right)$

Minors
\begin{eqnarray*} [M_{ij}] & = & \left[ \begin{array}{ccc}
\left|\begin{array}{cc} 5 & 6 \\ 1 & 1 \end{array}\right| &
\left|\begin{array}{cc} 4 & 6 \\ 1 & 1 \end{array}\right| &
\left|\begin{array}{cc} 4 & 5 \\ 1 & 1 \end{array}\right| \\ \\
\left|\begin{array}{cc} 2 & 3 \\ 1 & 1 \end{array}\right| &
\left|\begin{array}{cc} 1 & 3 \\ 1 & 1 \end{array}\right| &
\left|\begin{array}{cc} 1 & 2 \\ 1 & 1 \end{array}\right| \\ \\
\left|\begin{array}{cc} 2 & 3 \\ 5 & 6 \end{array}\right| &
\left|\begin{array}{cc} 1 & 3 \\ 4 & 6 \end{array}\right| &
\left|\begin{array}{cc} 1 & 2 \\ 4 & 5 \end{array}\right| \\
\end{array}\right]\\ & = & \left( \begin{array}{ccc} -1 & -2 & -1 \\
 -1 & -2 & -1 \\ -3 & -6 & -3\end{array} \right) \end{eqnarray*}

Cofactors

$[A_{ij}] =  \left( \begin{array}{ccc} -1 & 2 & -1
\\  1 & -2 & 1 \\ -3 & 6 & -3\end{array} \right)$



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