Express the determinant in the form ai + bj + ck for real numbers a, b, and c.

Question

Express the determinant in the form ai + bj + ck for real numbers a, b, and c.

| i j k

5 -8 -1

3 0 1 |

Answer 1

The determinant of a 3 by 3 matrix is equivalent to the top values times their determinants.

Remember the second value is negative.

D = i * M₁₁ - j*M₁₂ + k*M₁₃

Remember the minor is found by crossing out all numbers vertical and horizontal of the origin.

So for M₁₂ the determinant you need to calculate is

[5 -1

3 1]

M₁₂ = 5*1 - 3*-1 = 8

Substituting into the main determinant

-j*8 = bj

So b = -8

Answer 2

-8i-8j+24k, i|-8, -1, 0, 1| - j|5, -1, 3, 1| + k|5, -8, 3, 0|