Step by step
The question
Question: find det(A²), when det A = 3.
det A = 3
1 Use the rule
A to the power 2 is 2 As multiplied.
Each one brings its det A.
A² = A × A
det(AB) = det A × det B
By the product rule.
det(A2) = det A × det A
Careful!
Not 2 times det A: det A multiplied 2 times.
Careful: not 2 × 3 ✗
2 Work it out
That is 3 to the power 2.
det(A2) = 32
So det(A²) = 9.
det(A2) = 9
3 Check
Let's check with an easy A: 3 and 1 on the diagonal.
A = [NaN, 0; 0, 1]
Multiplying diagonal matrices multiplies their diagonals: 3 to the power 2, and 1.
A2 = [9, 0; 0, 1]
Its determinant: 9.
det(A2) = 9 × 1 = 9
We get 9 again.
So the answer is right!
Let's look back.
First: use the rule.
Then: work it out.
Last: check.
Answer
det(A2) = 9