The determinant of a product is the product of the determinants. Two area factors applied in turn compound.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Swapping rows
Swapping two rows negates the determinant. The transformation has been flipped over.
Scaling a row
Multiplying one row by k multiplies the determinant by k. Stretching in one direction stretches area by the same factor.
Adding a multiple
Adding a multiple of one row to another leaves the determinant unchanged. This is what makes elimination safe.
Triangular matrices
If everything below the diagonal is zero, the determinant is just the product of the diagonal entries.
Transpose
A matrix and its transpose have the same determinant. Every row rule therefore has a matching column rule.
Determinants multiply
det(AB) = det(A)det(B). Composing transformations multiplies their area factors, which makes this property geometrically obvious even though the algebraic proof is not.
How row operations affect it
Swapping rows negates it, scaling a row scales it, and adding a multiple of one row to another leaves it unchanged. Those rules make determinants computable by elimination.
Triangular matrices are easy
The determinant of a triangular matrix is the product of its diagonal entries. Since elimination produces a triangular form, that is the efficient route for any large matrix.
Transposing changes nothing
det(Aᵀ) = det(A), so every statement about rows has a matching statement about columns. That symmetry halves the number of facts to remember.
Step 2: Try It Yourself
Tap and try it out.
- Determinant of A20
A non-zero determinant means the matrix is invertible. Its size is the factor by which areas are scaled — here 20.
Step 3: Watch an Example
One step at a time.
Watch Leila Avoid the Arithmetic
Leila needs the determinant of a triangular matrix with diagonal 2, 5 and 3.
- Step 1
She checks that every entry below the diagonal is zero.
Step 4: Your Turn
Practice makes it stick.
The Product
Problem 1 of 2
Determinants 4 and 9. What is the determinant of the product matrix?
The Swap
Problem 2 of 2
A determinant of 7. Two rows are swapped. What is the new determinant?
Use the Rules
1 of 8
A triangular matrix with diagonal 3, 4. What is the determinant?
2 of 8
A determinant of 5. Row 1 is multiplied by 3. What is the new determinant?
3 of 8
A determinant of 8. Twice row 1 is added to row 2. What is the new determinant?
4 of 8
A determinant of 6. What is the determinant of the transpose?
5 of 8
Determinants 3 and 0. What is the determinant of the product?
6 of 8
A matrix with a whole row of zeros. What is its determinant?
7 of 8
Match each row operation to its effect on the determinant.
Tap a card on the left to start.
8 of 8
A triangular matrix with diagonal 2, 0, 9. What is the determinant?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A triangular matrix with diagonal 5, 2, 3. What is the determinant?
Question 2 of 2
Which row operation leaves the determinant unchanged?
What You Learned
- Determinants multiply across a product, because area factors compound.
- Swapping rows negates it, scaling a row scales it, and adding a multiple leaves it alone.
- A triangular determinant is just the product of the diagonal entries.