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# Evaluate the determinant: $\begin{vmatrix} 3&-1&-2 \\ 0&0&-1 \\3&-5&0 \end{vmatrix}$

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:(i) $Evaluate:\begin{vmatrix}3 & -1 & -2\\0 & 0 &-1\\3 & -5 & 0\end{vmatrix}$

We know to evaluate the value of the determinant of order $3\times 3$

Therefore $\mid A\mid=a_{11}(a_{22}\times a_{33}-a_{23}\times a_{32})-a_{12}(a_{21}\times a_{32}-a_{23}\times a_{31})+a_{13}(a_{21}\times a_{32}- a_{22}\times a_{31})$

$\mid A\mid=3[(0\times 0-(-1\times -5)]-(-1)[0\times 0-(-1\times 31]+(-2)[0\times 0-(0)\times 3]$

$\qquad=3(-5)+1(3)-2(0)$

$\qquad=-15+3-0$

$\qquad=-12$

edited Feb 20, 2013