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Properties of Determinants

Find the valu...

Question

Find the value of $∣ ∣ ∣ ∣ \begin{matrix} 67 & 19 & 21 39 & 13 & 14 81 & 24 & 26 \end{matrix} ∣ ∣ ∣ ∣$ .

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Solution

We have
$∣ ∣ ∣ ∣ \begin{matrix} 67 & 19 & 21 39 & 13 & 14 81 & 24 & 26 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} 10 + 57 & 19 & 21 0 + 39 & 13 & 14 9 + 72 & 24 & 26 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} 10 & 19 & 21 0 & 13 & 14 9 & 24 & 26 \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} 57 & 19 & 21 39 & 13 & 14 72 & 24 & 26 \end{matrix} ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ \begin{matrix} 10 & 19 & 21 0 & 13 & 14 9 & 24 & 26 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} 10 & 19 & 19 + 2 0 & 13 & 13 + 1 9 & 24 & 24 + 2 \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} 10 & 19 & 2 0 & 13 & 1 9 & 24 & 2 \end{matrix} ∣ ∣ ∣ ∣$
$= 10 ∣ ∣ ∣ \begin{matrix} 13 & 1 24 & 2 \end{matrix} ∣ ∣ ∣ + 9 ∣ ∣ ∣ \begin{matrix} 19 & 2 13 & 1 \end{matrix} ∣ ∣ ∣ = 20 - 63 = - 43$ .

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