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

Note: This is part 2 of a 4 part question, split as 4 separate questions here.

Toolbox:
• To evaluate a matrix of order $3\times 3$
• $\mid A\mid=\begin{vmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{vmatrix}$
• Therefore $\mid A\mid=a_{11}(a_{22}\times a_{33}-a_{23}\times a_{32})-a_{12}(a_{21}\times a_{32}-a_{23}\times a_{31})+a_{13}(a_{21}\times a_{32}-a_{22}\times a_{31})$
Given: $Evaluate:\begin{vmatrix}3 & -4 & 5\\1& 1 &-2\\2 & 3 & 1\end{vmatrix}$
We know to evaluate the value of the determinant of order $3\times 3$
Therefore $\mid A\mid=a_{11}(a_{22}\times a_{33}-a_{23}\times a_{32})-a_{12}(a_{21}\times a_{32}-a_{23}\times a_{31})+a_{13}(a_{21}\times a_{32}- a_{22}\times a_{31})$
$\mid A\mid=3[(1\times 1-(3\times -2)]-(-4)[(1\times 1)-(-2\times 2]+5[(1\times 3)-(1\times 2)]$
$\qquad=3(1+6)+4(1+4)+52(3-2)$
$\qquad=21+20+5$
$\qquad=46$
edited Feb 18, 2014