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

Jun 2, 2016

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

#### Explanation:

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

$\textcolor{w h i t e}{\text{XXX}} = + \left(- 3\right) \textcolor{red}{| \left(- 8 , 1\right) , \left(3 , 9\right) |} - \left(2\right) \textcolor{b l u e}{| \left(5 , 1\right) , \left(6 , 9\right) |} + \left(7\right) \textcolor{g r e e n}{| \left(5 , - 8\right) , \left(6 , 3\right) |}$

$\textcolor{red}{| \left(- 8 , 1\right) , \left(3 , 9\right) | = + \left(\left(- 8\right) \times 9\right) - \left(1 \times 3\right) = - 72 - 3 = - 75}$

$\textcolor{b l u e}{| \left(5 , 1\right) , \left(6 , 9\right) | = + \left(5 \times 9\right) - \left(1 \times 6\right) = 45 - 6 = 39}$

$\textcolor{g r e e n}{| \left(5 , - 8\right) , \left(6 , 3\right) | = + \left(5 \times 3\right) - \left(\left(- 8\right) \times 6\right) = 15 + 48 = 63}$

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

$\textcolor{w h i t e}{\text{XXX")=(-3)xxcolor(red)(} \left(- 75\right)} - 2 \times \textcolor{b l u e}{39} + 7 \times \textcolor{g r e e n}{63}$

$\textcolor{w h i t e}{\text{XXX}} = 225 - 78 + 441$

$\textcolor{w h i t e}{\text{XXX}} = 588$

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Note that in real life no-one calculate a determinant by hand like this.