Step by step
The question
Question: find det(A³), when det A = 2.
det A = 2
1 Use the rule
A to the power 3 is 3 As multiplied.
Each one brings its det A.
A³ = A × A × A
det(AB) = det A × det B
By the product rule.
det(A3) = det A × det A × det A
Careful!
Not 3 times det A: det A multiplied 3 times.
Careful: not 3 × 2 ✗
2 Work it out
That is 2 to the power 3.
det(A3) = 23
So det(A³) = 8.
det(A3) = 8
3 Check
Let's check with an easy A: 2 and 1 on the diagonal.
A = [NaN, 0; 0, 1]
Multiplying diagonal matrices multiplies their diagonals: 2 to the power 3, and 1.
A3 = [8, 0; 0, 1]
Its determinant: 8.
det(A3) = 8 × 1 = 8
We get 8 again.
So the answer is right!
Let's look back.
First: use the rule.
Then: work it out.
Last: check.
Answer
det(A3) = 8