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{\bf Question}

Evaluate the determinant $|A|$ associated with the matrix $A =
\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 1 & 1
\end{array} \right)$ in the following ways:
\begin{description}
\item[(a)] Using the cofactor formula on the third row;
\item[(b)] Using the cofactor formula on the second column;
\item[(c)] By subtracting the second row from the first row, and
then using some general properties of determinants.
\end{description}

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{\bf Answer}

$\left| \begin{array}{ccc} 1& 2 & 3 \\ 4& 5 & 6 \\ 1 & 1 &1
\end{array} \right|$
\begin{description}
\item[(a)]
Using cofactor formula on the third row

$\det(A) = 1 \times (-3) + 1 \times 6 + 1 \times(-3) = 0$
\item[(b)]
Using cofactor formula on the second colunm

$\det(A) = 2 \times 2 + 5 \times (-2) + 1 \times 6 = 0$
\item[(c)]
general properties (row 2 $\rightarrow$ row 3 - row 1)

 $\det(A) = \left| \begin{array}{ccc} 1& 2 & 3
\\ 3& 3 & 3 \\ 1 & 1 &1 \end{array} \right| = 3\left| \begin{array}{ccc} 1& 2 & 3
\\ 1 & 1 & 1 \\ 1 & 1 &1 \end{array} \right| = 0$ because two rows
are identical.
\end{description}


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