Question

How do you find the determinant of `((12, 5, -2), ( -3, 0, 1), (-5, 4, 2))`?

Answer 1

`|(12,5,-2),(-3,0,1),(-5,4,2)|=color(green)(-19)`

Explanation:

Method 1 (what your teacher probably wanted)
`|(color(blue)(12),color(blue)(5),color(blue)(-2)),(-3,0,1),(-5,4,2)|`

`=color(red)(+)color(blue)(12)|(color(purple)(0),color(green)(1)),(color(green)(4),color(purple)(2))|color(red)(-)color(blue)(5)|(color(purple)(-3),color(green)(1)),(color(green)(-5),color(purple)(2))|color(red)(+)color(blue)(""(-2))|(color(purple)(-3),color(green)(0)),(color(green)(-5),color(purple)(4))|`

`=color(blue)(12)(color(purple)(0xx2)-color(green)(4xx1))`
`color(white)("X")-color(blue)(5)(color(purple)((-3)xx2)-color(green)((-5)xx1))`
`color(white)("X")color(blue)(-2)(color(purple)((-3)xx4)-color(green)((-5)xx0))`

`=color(blue)(12)(color(orange)(-4))-color(blue)(5)(color(orange)(-1))color(blue)(-2)(color(orange)(-12))`

`=-48+5+24`

`=-19`

You might have been expected to exchange a pair of columns (or a pair of rows) to reduce the calculations a bit. Just remember:
`color(white)("XXX")|(12,5,-2),(-3,0,1),(-5,4,2)| = color(red)(-)|(-3,0,1),(12,5,2),(-5,4,2)|`
(exchanging rows or columns negates the value of the determinant).

Method 2 (how you would do this if you needed it in real life)
Enter the matrix into a spreadsheet and use a builtin function: