Step by step
The question
Question: find det(AB), when det A = 3 and det B = −2.
det A = 3, det B = −2
1 Use the rule
The determinant of a product is the product of the determinants.
det(AB) = det A × det B
So det(AB) is det A times det B.
det(AB) = det A × det B
Careful!
Multiply the two determinants.
Do not add them.
Careful: multiply the determinants, do not add them
2 Work it out
3 times (−2).
det(AB) = 3 × (−2)
So det(AB) = −6.
det(AB) = −6
3 Check
Let's check with easy matrices: A with 3 and 1 on the diagonal, B with −2 and 1.
A = [NaN, 0; 0, 1], B = [NaN, 0; 0, 1]
Times two diagonal matrices: the diagonals multiply, 3 × (−2) and 1 × 1.
AB = [−6, 0; 0, 1]
Its determinant: −6.
det (AB) = −6 × 1 = −6
We get −6 again.
So the answer is right!
Let's look back.
First: use the rule.
Then: work it out.
Last: check.
Answer
det(AB) = −6