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Question
If \[A = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix} \text{ and } B = \begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}\] , verify that |AB| = |A| |B|.
Solution
\[\text{ Consider LHS}\]
\[AB = \begin{bmatrix}2 & 5 \\ 2 & 1\end{bmatrix}\begin{bmatrix}4 & - 3 \\ 2 & 5\end{bmatrix}\]
\[ = \begin{bmatrix}8 + 10 & - 6 + 25 \\ 8 + 2 & - 6 + 5\end{bmatrix} = \begin{bmatrix}18 & 19 \\ 10 & - 1\end{bmatrix}\]
\[\left| AB \right| = - 18 - 190 = - 208\]
\[\text{ Consider RHS}\]
\[\left| A \right| = 2 - 10 = - 8\]
\[\left| B \right| = 20 - \left( - 6 \right) = 26\]
\[\left| A \right|\left| B \right| = - 8 \times 26 = - 208\]
\[ \therefore\text{ LHS = RHS }\]
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