Evaluate the determinant: 3&-4&5 1&1&-2 2&3&1 - Vidyadip

Question

Evaluate the determinant:$ \begin{vmatrix}3&-4&5\\1&1&-2\\2&3&1\end{vmatrix}$

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Answer

Let $\text{A}=\begin{vmatrix}3&-4&5\\1&1&-2\\2&3&1\end{vmatrix}.$ By expanding along the first row, we have: $|\text{A}|=3\begin{vmatrix}1&-2\\3&1\end{vmatrix}+4\begin{vmatrix}1&-2\\2&1\end{vmatrix}+5\begin{vmatrix}1&1\\2&3\end{vmatrix}$$=3(1+6)+4(1+4)+5(3-2)$
$=3(7)+4(5)+5(1)$
$=21+20+5=46$

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