Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 20/Solution 1

We compute

${\begin{aligned}{\begin{vmatrix}1&3&2&0&1\\2&7&1&0&2\\1&5&6&-1&7\\0&0&3&0&0\\1&1&4&0&4\\\end{vmatrix}}&=-3\cdot {\begin{vmatrix}1&3&0&1\\2&7&0&2\\1&5&-1&7\\1&1&0&4\\\end{vmatrix}}&&{\text{expanding row 4}}\\&=-3\cdot -1\cdot {\begin{vmatrix}1&3&1\\2&7&2\\1&1&4\end{vmatrix}}&&{\text{expanding column 3}}\\&=-3\cdot -1\cdot \left({\begin{vmatrix}7&2\\1&4\end{vmatrix}}-3\cdot {\begin{vmatrix}2&2\\1&4\end{vmatrix}}+{\begin{vmatrix}2&7\\1&1\end{vmatrix}}\right)&&{\text{expanding row 1}}\\&=-3\cdot -1\cdot ((7\cdot 4-2\cdot 1)-3(2\cdot 4-2\cdot 1)+(2\cdot 1-7\cdot 1))\\&=\color {blue}9.\end{aligned}}$