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{\bf Question}

Calculate the determinants of the following matrices.

$$A = \left( \begin{array}{rrr} 1 & {-1} & 0 \\ 1 & {-1} & 1 \\ 2
& 3 & 1 \end{array} \right);\ B = \left( \begin{array}{rrr} 1 & 0
& 0 \\ 0 & {-2} & 1 \\ 3 & {-1} & {-1} \end{array} \right). $$

Calculate the determinants  of the matrix $AB$ and verify that it
is equal to $\det(A)\det(B)$.



{\bf Answer}

$$\det(A)=\left | \begin{array} {rrr} {1} & {-1} & {0}\\ {1} &
{-1} & {1}\\ {2} & {3} & {1} \end{array} \right|=(1)\left |
\begin{array} {rr} {-1} & {1}\\ {3} & {1} \end{array} \right|-(-1)\left |
\begin{array} {rr} {1} & {1}\\ {2} & {1} \end{array} \right|+0=(-1-3)+(1-2)=-5$$

$$\det(B)=\left | \begin{array} {rrr} {1} & {0} & {0}\\ {0} & {-2}
& {1}\\ {3} & {-1} & {-1} \end{array} \right|=(1)\left |
\begin{array} {rr} {-2} & {1}\\ {-1} & {-1} \end{array} \right|-0+0=(2+1)=3$$

$$AB=\left (\begin{array} {rrr} {1} & {2} & {-1}\\ {4} & {1} &
{-2}\\ {5} & {-7} & {2} \end{array} \right)$$

so that

\begin{eqnarray*} \det(AB) =\left | \begin{array} {rrr} {1} & {2} & {-1}\\ {4} &
{1} & {-2}\\ {5} & {-7} & {2} \end{array} \right| & = & (1)\left |
\begin{array} {rr} {1} & {-2}\\ {-7} & {2} \end{array} \right|-(2)\left |
\begin{array} {rr} {4} & {-2}\\ {5} & {2} \end{array} \right|+(-1)\left |
\begin{array} {rr} {4} & {1}\\ {5} & {-7} \end{array} \right|\\ & =
& (2-14)-(2)(8+10)-(-28-5)\\ & = & -15\\ & = & (-5)(3)\\ & = &
\det (A) \det (B). \end{eqnarray*}



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