Properties of Determinants

Rules that let you avoid most of the arithmetic.

12 problems · about 23 minutes · N-VM.C.12

What this lesson teaches

The student applies determinant properties to compute and reason without full expansion.

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.

Straightforward practice. Get the method working first.

5 problems

A triangular matrix with diagonal 5, 2, 3. What is the determinant?
Answer 30
Why 30.
Which row operation leaves the determinant unchanged?
Answer Adding a multiple of one row to another.
Why Adding a multiple.
A triangular matrix with diagonal 3, 4. What is the determinant?
Answer 12
Why Multiply the diagonal.
A determinant of 5. Row 1 is multiplied by 3. What is the new determinant?
Answer 15
Why The factor carries through.
A determinant of 8. Twice row 1 is added to row 2. What is the new determinant?
Answer 8
Why This operation changes nothing.

The same ideas with more to keep track of.

3 problems

A determinant of 6. What is the determinant of the transpose?
Answer 6
Why Transposing preserves it.
Determinants 3 and 0. What is the determinant of the product?
Answer 0
Why Anything times zero.
A matrix with a whole row of zeros. What is its determinant?
Answer 0
Why Scaling that row by 0 changed nothing.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each row operation to its effect on the determinant.
Answer Swap two rows → The determinant changes sign; Multiply a row by k → The determinant is multiplied by k; Add a multiple of one row to another → The determinant is unchanged
Why Only one of the three leaves it alone.
A triangular matrix with diagonal 2, 0, 9. What is the determinant?
Answer 0
Why A zero on the diagonal.
The Product: Determinants 4 and 9. What is the determinant of the product matrix?
Answer 36
Why 36.
The Swap: A determinant of 7. Two rows are swapped. What is the new determinant?
Answer -7
Why −7.