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

$\begin{array}{1 1} -19 \\ 19 \\ 18 \\ -18 \end{array}$

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:(iv) $\mid A\mid=\begin{vmatrix}2 & -1 & -2\\0& 2 &-1\\3 & -5 & 0\end{vmatrix}$

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

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

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

$\qquad=2(-5)+1(3)+2(-6)$

$\qquad=-10+3-12$

$\qquad=-19$
edited Apr 30, 2014