Lesson 2: Properties of Determinants
Estimated time: 35-40 minutes
Learning Objectives
- Describe how each row operation affects the determinant
- Use row reduction to compute determinants efficiently
- Apply the multiplicative property: det(AB) = det(A)det(B)
- State additional properties: det(A^T) = det(A), det(A^{-1}) = 1/det(A)
Row Operations and the Determinant
Three Rules:
- Row swap: Swapping two rows negates the determinant. det(E_swap A) = -det(A).
- Row scaling by c: Multiplying a row by c multiplies the determinant by c. det(E_scale A) = c * det(A).
- Row replacement: Adding a multiple of one row to another does not change the determinant. det(E_replace A) = det(A).
Example: Computing det via Row Reduction
A = [2 4; 1 5]. Instead of the formula, let us row reduce.
R_1 ↔ R_2 (swap -- det changes sign): [1 5; 2 4]. Det of new = -det(A).
R_2 - 2R_1 (replacement -- no change): [1 5; 0 -6]. Det = -det(A).
Upper triangular: det = product of diagonal = 1*(-6) = -6. So -det(A) = -6, meaning det(A) = 6.
Check with formula: 2(5) - 4(1) = 10 - 4 = 6. Confirmed!
The Multiplicative Property
det(AB) = det(A) * det(B) for any two n x n matrices A and B.
Example
A = [2 0; 0 3], det(A) = 6. B = [1 1; 0 2], det(B) = 2.
AB = [2 2; 0 6], det(AB) = 12 = 6 * 2 = det(A)*det(B). ✓
Warning
det(A + B) is NOT equal to det(A) + det(B) in general. The determinant is multiplicative, not additive.
More Properties
Additional Properties:
- det(A^T) = det(A)
- det(A^{-1}) = 1/det(A) (when A is invertible)
- det(cA) = c^n det(A) for an n x n matrix (every row is scaled by c)
- If A has a row or column of all zeros, det(A) = 0
- If two rows (or columns) are identical, det(A) = 0
- If two rows (or columns) are proportional, det(A) = 0
Check Your Understanding
1. If det(A) = 5 and det(B) = -3, what is det(AB)?
2. If A is 3x3 and det(A) = 4, what is det(2A)?
3. You row reduce a 3x3 matrix using 1 row swap and no scaling, arriving at an upper triangular matrix with diagonal entries 1, 3, -2. What is the original determinant?
4. True or False: det(A + B) = det(A) + det(B).
Key Takeaways
- Row swap negates det; row scaling by c multiplies det by c; row replacement leaves det unchanged
- det(AB) = det(A)det(B) (multiplicative property)
- det(A^T) = det(A) and det(A^{-1}) = 1/det(A)
- det(cA) = c^n det(A) for n x n matrix A
- A row/column of zeros, or two identical/proportional rows, forces det = 0