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Adjoint of a Matrix

If A=[ 2 5;...

Question

If $A = [\begin{matrix} 2 & 5 2 & 1 \end{matrix}]$ and $B = [\begin{matrix} 4 & - 3 2 & 5 \end{matrix}]$ , verify that $| A B | = | A | | B | .$

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Solution

$A B = ∣ ∣ ∣ \begin{matrix} 2 & 5 2 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 4 & - 3 2 & 5 \end{matrix} ∣ ∣ ∣$
$= ∣ ∣ ∣ \begin{matrix} 8 + 10 & - 6 + 25 8 + 2 & - 6 + 5 \end{matrix} ∣ ∣ ∣$
$= ∣ ∣ ∣ \begin{matrix} 18 & 19 10 & - 1 \end{matrix} ∣ ∣ ∣$
$L . H . S = | A B | = - 18 - 190 = - 208$
$| A | = ∣ ∣ ∣ \begin{matrix} 2 & 5 2 & 1 \end{matrix} ∣ ∣ ∣$
$= 2 - 10 = - 8$
$| B | = ∣ ∣ ∣ \begin{matrix} 4 & - 3 2 & 4 \end{matrix} ∣ ∣ ∣$
$= 20 + 6 = 26$
$R . H . S = | A | | B | = - 8 \times 26 = - 208$
$∴$ L.H.S $=$ R.H.S
Hence verified.

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