Minor, Cofactor, Adjoint, and Inverse
Minor Matrix
The minor of an element in the ith row and jth column of a square matrix A is denoted by Mij and is defined as the determinant of the submatrix formed by removing the ith row and jth column from A:
Mij = |Aij|
Here, Aij is the submatrix obtained by deleting the ith row and jth column of A, and |Aij| denotes its determinant.
Cofactor Matrix
The cofactor of an element in the ith row and jth column of A is defined as:
Cij = (-1)i+j · Mij
Where Mij is the minor of the element. The factor (-1)i+j ensures the correct sign for cofactor expansion.
Calculating Determinant Using Minors and Cofactors
To calculate the determinant of a square matrix A using minors and cofactors, follow these steps:
- Choose an element: Pick an element \( a_{ij} \) from any row or column of \( A \).
- Form the submatrix: Remove the ith row and jth column from \( A \) to obtain the submatrix \( A_{ij} \).
- Calculate the minor: The minor \( M_{ij} \) is the determinant of the submatrix: \[ M_{ij} = |A_{ij}| \]
- Compute the cofactor: Multiply the minor by the sign factor: \[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \] The sign alternates in a checkerboard pattern across the matrix.
- Expand along a row or column: The determinant of \( A \) is the sum of the products of elements and their cofactors along the chosen row or column: \[ \det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + \dots + a_{in}C_{in} \]
Example: 3 × 3 Matrix
Consider the matrix:
\[ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]Calculate the minors for the first row:
\[ \begin{aligned} M_{11} &= \begin{vmatrix} e & f \\ h & i \end{vmatrix} = ei - fh, \\ M_{12} &= \begin{vmatrix} d & f \\ g & i \end{vmatrix} = di - fg, \\ M_{13} &= \begin{vmatrix} d & e \\ g & h \end{vmatrix} = dh - eg \end{aligned} \]Calculate the cofactors:
\[ \begin{aligned} C_{11} &= (+1) \cdot M_{11} = ei - fh, \\ C_{12} &= (-1) \cdot M_{12} = fg - di, \\ C_{13} &= (+1) \cdot M_{13} = dh - eg \end{aligned} \]Finally, expand along the first row to find the determinant:
\[ \det(A) = a C_{11} + b C_{12} + c C_{13} = a(ei - fh) + b(fg - di) + c(dh - eg) \]Adjoint (Adjugate)
The adjoint (or adjugate) of a square matrix A is the transpose of its cofactor matrix:
Adj(A) = [Cij]T
Inverse of a Matrix
If A is invertible (i.e., det(A) ≠ 0), its inverse is given by:
A-1 = Adj(A) / det(A)
Inverse of a Product
The inverse of a product of two invertible matrices is the product of their inverses in reverse order:
(AB)-1 = B-1 A-1
Proof:
Start with the identity: (AB)(AB)-1 = I
Pre-multiply by A-1: A-1(AB)(AB)-1 = A-1 ⇒ B(AB)-1 = A-1
Pre-multiply by B-1: B-1B(AB)-1 = B-1A-1 ⇒ (AB)-1 = B-1A-1
Adjoint of a Product
The adjoint of a product of two matrices equals the product of their adjoints in reverse order:
Adj(AB) = Adj(B) · Adj(A)
Proof:
From the inverse formula: (AB)-1 = Adj(AB) / det(AB) and (AB)-1 = B-1A-1.
Also, det(AB) = det(A) · det(B), and A-1 = Adj(A)/det(A), B-1 = Adj(B)/det(B).
Multiplying accordingly gives: Adj(B) · Adj(A) = det(A) · det(B) · B-1 · A-1, so
Adj(AB) = Adj(B) · Adj(A)