Lesson 1: Determinant -- Definition and Cofactor Expansion
Estimated time: 35-45 minutes
Learning Objectives
- Define the determinant of a square matrix
- Compute 2x2 and 3x3 determinants directly
- Define minors and cofactors
- Apply cofactor expansion along any row or column
The 2x2 Determinant
2x2 Determinant: For A = [a b; c d], det(A) = ad - bc.
Worked Example
det([3 1; 2 5]) = 3(5) - 1(2) = 15 - 2 = 13.
det([4 6; 2 3]) = 4(3) - 6(2) = 12 - 12 = 0. (This matrix is singular.)
Minors and Cofactors
Minor M_{ij}: The determinant of the (n-1)x(n-1) matrix obtained by deleting row i and column j from A.
Cofactor C_{ij}: C_{ij} = (-1)^{i+j} M_{ij}. The sign alternates in a checkerboard pattern: + - + / - + - / + - + ...
Example: Sign Pattern for 3x3
[ - + - ]
[ + - + ]
C_{11} = +M_{11}, C_{12} = -M_{12}, C_{13} = +M_{13}, etc.
Cofactor Expansion
Cofactor Expansion along row i: det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + ... + a_{in}C_{in}.
You can also expand along any column. The result is the same regardless of which row or column you choose.
Worked Example: 3x3 Determinant
Find det(A) where A = [2 1 3; 0 -1 2; 1 4 -1].
Expand along row 1:
det(A) = 2 * C_{11} + 1 * C_{12} + 3 * C_{13}
C_{11} = +det([-1 2; 4 -1]) = +( (-1)(-1) - (2)(4) ) = 1 - 8 = -7
C_{12} = -det([0 2; 1 -1]) = -( (0)(-1) - (2)(1) ) = -(-2) = 2
C_{13} = +det([0 -1; 1 4]) = +( (0)(4) - (-1)(1) ) = 0 + 1 = 1
det(A) = 2(-7) + 1(2) + 3(1) = -14 + 2 + 3 = -9.
Strategy: Choose Wisely
Expand along the row or column with the most zeros to minimize computation. Each zero eliminates one cofactor calculation.
Example: Expanding Along a Column with Zeros
For A = [2 1 3; 0 -1 2; 1 4 -1], expand along column 1:
det(A) = 2*C_{11} + 0*C_{21} + 1*C_{31}
The 0 in position (2,1) means we skip that cofactor entirely.
C_{11} = +det([-1 2; 4 -1]) = -7 (computed above)
C_{31} = +det([1 3; -1 2]) = 2 - (-3) = 5
det(A) = 2(-7) + 0 + 1(5) = -14 + 5 = -9. Same answer!
The Determinant and Invertibility
Key Theorem: A square matrix A is invertible if and only if det(A) is not equal to 0.
This connects determinants to everything we learned in Modules 1-2: invertibility, unique solutions, and pivot columns.
Check Your Understanding
1. Compute det([5 2; 3 1]).
2. What is the cofactor C_{23} of a 3x3 matrix?
3. Compute det([1 0 0; 3 2 0; 1 5 4]).
4. If det(A) = 0, what can you conclude?
Key Takeaways
- 2x2 determinant: det([a b; c d]) = ad - bc
- Cofactor C_{ij} = (-1)^{i+j} times the minor M_{ij}
- Cofactor expansion works along any row or column -- choose the one with the most zeros
- Triangular matrices: det = product of diagonal entries
- A is invertible if and only if det(A) is not zero