# How do you find the determinant of |(4,-1,-2), (0, 2, 1), (2,1,3)|?

Jan 12, 2017

The answer is $= 26$

#### Explanation:

One method is as follows :

$| \left(a , b , c\right) , \left(d , e , f\right) , \left(g , h , i\right) |$

$= a | \left(e , f\right) , \left(h , i\right) | - b | \left(d , f\right) , \left(g , i\right) | + c | \left(d , e\right) , \left(g , h\right) |$

$= a \left(e i - f h\right) - b \left(\mathrm{di} - g f\right) + c \left(\mathrm{dh} - e g\right)$

Therefore,

$| \left(4 , - 1 , - 2\right) , \left(0 , 2 , 1\right) , \left(2 , 1 , 3\right) |$

$= 4 \cdot | \left(2 , 1\right) , \left(1 , 3\right) | + 1 \cdot | \left(0 , 1\right) , \left(2 , 3\right) | - 2 \cdot | \left(0 , 2\right) , \left(2 , 1\right) |$

$= 4 \cdot \left(6 - 1\right) + 1 \cdot \left(0 - 2\right) - 2 \cdot \left(0 - 4\right)$

$= 4 \cdot 5 - 1 \cdot 2 - 2 \cdot - 4$

$= 20 - 2 + 8$

$= 26$