Determinant of a Product
The determinant of a product is the product of determinants.
Formula
Variables
Example
Multiplicative property
How it works
For square matrices of the same size, multiplying matrices composes linear transformations, and the determinant measures the scaling factor of that transformation. Since doing A and then B scales volumes by det(A) and det(B) separately, the combined effect scales by their product, which is why det(AB) = det(A)det(B). This property follows from the determinant’s multilinearity and how matrix multiplication combines columns or rows through linear combinations.
When to use it
Use this when you need the determinant of a matrix product without multiplying the matrices out first. It is especially useful in proofs, invertibility arguments, eigenvalue problems, and when tracking how a sequence of linear transformations changes area or volume.
Worked example, step by step
Let A = [[1, 2], [3, 4]] and B = [[0, 5], [1, -1]]. Step 1: Compute det(A) = (1)(4) - (2)(3) = 4 - 6 = -2. Step 2: Compute det(B) = (0)(-1) - (5)(1) = 0 - 5 = -5. Step 3: Use the product rule: det(AB) = det(A)det(B) = (-2)(-5) = 10. Step 4: Check directly by multiplying: AB = [[(1)(0)+(2)(1), (1)(5)+(2)(-1)], [(3)(0)+(4)(1), (3)(5)+(4)(-1)]] = [[2, 3], [4, 11]]. Step 5: det(AB) = (2)(11) - (3)(4) = 22 - 12 = 10. Final answer: det(AB) = 10.
Common mistakes to avoid
- Assuming the rule works for non-square matrices
- Forgetting that det(AB) = det(A)det(B) does not mean AB = det(A)det(B)
- Reversing the order of multiplication and thinking det(AB) depends on whether A and B commute
Did You Know?
A zero determinant signals a singular matrix that has no inverse and squashes space into a lower dimension.
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