# How do you find the determinant of  ((5, 1, 4), (-1, f, e), (g, 0 , 1))?

Nov 2, 2016

$| \left(5 , 1 , 4\right) , \left(- 1 , f , e\right) , \left(g , 0 , 1\right) | = 5 f + e + g - 4 f g$

#### Explanation:

Expanding using row1 we have;

$| \left(5 , 1 , 4\right) , \left(- 1 , f , e\right) , \left(g , 0 , 1\right) | = \left(5\right) | \left(f , e\right) , \left(0 , 1\right) | - \left(1\right) | \left(- 1 , 1\right) , \left(g , e\right) | + \left(4\right) | \left(- 1 , f\right) , \left(g , 0\right) |$

$= \left(5\right) \left\{\left(f\right) \left(1\right) - \left(0\right) \left(e\right)\right\} - \left(1\right) \left\{\left(- 1\right) \left(e\right) - \left(g\right) \left(1\right)\right\} + \left(4\right) \left\{\left(- 1\right) \left(0\right) - \left(g\right) \left(f\right)\right\}$
$= \left(5\right) \left(f\right) - \left(1\right) \left(- e - g\right) + \left(4\right) \left(- f g\right)$
$= 5 f + e + g - 4 f g$