Question

If $A=\left[\begin{array}{cc}2& 5\\ 2& 1\end{array}\right]\mathrm{and}B=\left[\begin{array}{cc}4& -3\\ 2& 5\end{array}\right]$, verify that |AB| = |A| |B|.

Answer 1

$$\begin{gathered} \mathrm{Consider}\mathrm{LHS} \\ AB=\left[\begin{array}{cc}2& 5\\ 2& 1\end{array}\right]\left[\begin{array}{cc}4& -3\\ 2& 5\end{array}\right] \\ =\left[\begin{array}{cc}8+10& -6+25\\ 8+2& -6+5\end{array}\right]=\left[\begin{array}{cc}18& 19\\ 10& -1\end{array}\right] \\ \left|AB\right|=-18-190=-208 \\ \mathrm{Consider}\mathrm{RHS} \\ \left|A\right|=2-10=-8 \\ \left|B\right|=20-\left(-6\right)=26 \\ \left|A\right|\left|B\right|=-8\times 26=-208 \\ \therefore \mathrm{LHS}=\mathrm{RHS} \end{gathered}$$