Minor of an Element
The minor of element aᵢⱼ is the determinant of the sub-matrix obtained by deleting the i-th row and j-th column from the matrix.
For a 3×3 matrix A with element at position (i, j), we delete that row and column, and the determinant of the remaining 2×2 matrix is Mᵢⱼ.
The minor is always a determinant (a number), not a matrix. For a 3×3 matrix, each minor is a 2×2 determinant.
M₁₁ = |5,6;8,9| = 45 − 48 = −3
Delete R₁ and C₂: submatrix [[4,6],[7,9]] → |4,6;7,9| = 36−42 = −6
Delete R₂ and C₁: submatrix [[2,3],[8,9]] → |2,3;8,9| = 18−24 = −6
Cofactor of an Element
The cofactor of element aᵢⱼ is the minor with an appropriate sign determined by position:
Aᵢⱼ = (−1)^(i+j) × Mᵢⱼ
The sign depends on whether (i + j) is even (+) or odd (−). This creates the checkerboard sign pattern:
Sign pattern for 3×3 (−1)^(i+j)
(−1)^(1+1) × M₁₁ = (+1)(−3) = −3
(−1)^(1+2) × M₁₂ = (−1)(−6) = +6
M₁₃ = |4,5;7,8| = 32−35 = −3; (−1)^(1+3)(−3) = (+1)(−3) = −3
Determinant Expansion via Cofactors: |A| = a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃ = sum of (element × its cofactor) along any row or column.
Adjoint (Adjugate) of a Matrix
The adjoint (or adjugate) of a matrix A is the transpose of the cofactor matrix.
adj(A) = [Aᵢⱼ]ᵀ
Where Aᵢⱼ is the cofactor of element aᵢⱼ. Each element of adj(A) at position (i,j) is the cofactor Aⱼᵢ (note the swap!).
Step-by-step to find adj(A):
Find ALL 9 cofactors (for a 3×3 matrix)
Write them as the cofactor matrix C
Transpose C → this gives adj(A)
(−1)²|0,1;2,1| = (0−2) = −2
(−1)³|1,1;−1,1| = −(1+1) = −2
(−1)⁴|1,0;−1,2| = (2−0) = 2
(−1)³|1,3;2,1| = −(1−6) = 5
(−1)⁴|2,3;−1,1| = (2+3) = 5
(−1)⁵|2,1;−1,2| = −(4+1) = −5
(−1)⁴|1,3;0,1| = (1−0) = 1
(−1)⁵|2,3;1,1| = −(2−3) = 1
(−1)⁶|2,1;1,0| = (0−1) = −1
adj(A) = Transpose of cofactor matrix:
adj(A) = [[−2, 5, 1],[−2, 5, 1],[2, −5, −1]]
Key Result: A · adj(A) = |A| · I
A · adj(A) = adj(A) · A = |A| · I
Where I is the identity matrix. This forms the foundation for finding the inverse of a matrix.
This result tells us:
- • If we multiply a matrix by its adjoint, we get a scalar multiple of the identity
- • The scalar is the determinant |A|
- • This directly leads to the inverse formula: A⁻¹ = adj(A) / |A|
| Term | Meaning | How to Get It |
|---|---|---|
| Mᵢⱼ | Minor of aᵢⱼ | Delete row i & col j → determinant |
| Aᵢⱼ or Cᵢⱼ | Cofactor of aᵢⱼ | (−1)^(i+j) × Mᵢⱼ |
| adj(A) | Adjoint / Adjugate | Transpose of cofactor matrix |