Step by step
The question
Question: find det(BA), when det A = 4 and det B = 5.
det A = 4, det B = 5
1 Use the rule
The determinant of a product is the product of the determinants.
det(AB) = det A × det B
So det(BA) is det B times det A, the same thing.
det(BA) = 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
4 times 5.
det(BA) = 4 × 5
So det(BA) = 20.
det(BA) = 20
3 Check
Let's check with easy matrices: A with 4 and 1 on the diagonal, B with 5 and 1.
A = [NaN, 0; 0, 1], B = [NaN, 0; 0, 1]
Times two diagonal matrices: the diagonals multiply, 4 × 5 and 1 × 1.
BA = [20, 0; 0, 1]
Its determinant: 20.
det (BA) = 20 × 1 = 20
We get 20 again.
So the answer is right!
Let's look back.
First: use the rule.
Then: work it out.
Last: check.
Answer
det(BA) = 20