Question

How do you find the determinant of `((-3, 2, 7), (5, -8, 1), (6, 3, 9))`?

Answer 1

#| (-3,2,7),(5,-8,1),(6,3,9) |=color(purple)(558)#

Explanation:

#| (-3,2,7),(5,-8,1),(6,3,9) |#

`color(white)("XXX")=+(-3)color(red)(|(-8,1),(3,9)|)-(2)color(blue)(|(5,1),(6,9)|)+(7)color(green)(|(5,-8),(6,3)|)`

`color(red)(|(-8,1),(3,9)|=+((-8)xx9)-(1xx3)=-72-3=-75)`

`color(blue)(|(5,1),(6,9)|=+(5xx9)-(1xx6)=45-6=39)`

`color(green)(|(5,-8),(6,3)|=+(5xx3)-((-8)xx6)=15+48=63)`

Therefore:
#| (-3,2,7),(5,-8,1),(6,3,9)|#

`color(white)("XXX")=(-3)xxcolor(red)(""(-75))-2xxcolor(blue)(39)+7xxcolor(green)(63)`

`color(white)("XXX")=225-78+441`

`color(white)("XXX") =588`

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Note that in real life no-one calculate a determinant by hand like this.
Instead a spreadsheet (or possibly a scientific calculator) would be used: