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

1 Answer
Aug 20, 2016

Answer:

#|(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:
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