Step by step
The question
Question: find det(A inverse), when det A = 4.
det A = 4
1 Use the rule
A times its inverse is I, and the determinant of I is 1.
A × A−1 = I, det I = 1
So det A times det of A inverse is 1, by the product rule.
det A × det(A−1) = 1
We put in det A = 4.
4 × det(A−1) = 1
2 Work it out
To remove times 4, we divide both sides by 4.
det(A−1) = 14
Careful!
The inverse turns D into 1 over D, not minus D.
Careful: not −4 ✗
So det(A inverse) = 1/4.
det(A−1) = 14
3 Check
Let's check with an easy A: 4 and 1 on the diagonal.
Its determinant is 4.
A = [NaN, 0; 0, 1]
Its inverse divides each diagonal number into 1: 1/4, and 1.
A−1 = [1 over 4, 0; 0, 1]
Its determinant: 1/4.
det(A−1) = 14 × 1 = 14
We get 1/4 again.
So the answer is right!
Let's look back.
First: use the rule.
Then: work it out.
Last: check.
Answer
det(A−1) = 14