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{\bf Question}

Calculate the determinants of the following matrices.

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

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



{\bf Answer}

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

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

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

so that

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



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