Minor, Cofactor, Adjoint, and Inverse

Minor Matrix

The minor of an element in the i^th row and j^th column of a square matrix A is denoted by Mij and is defined as the determinant of the submatrix formed by removing the i^th row and j^th column from A:

Mij = |Aij|

Here, Aij is the submatrix obtained by deleting the i^th row and j^th column of A, and |Aij| denotes its determinant.

Cofactor Matrix

The cofactor of an element in the i^th row and j^th 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:

1. Choose an element: Pick an element \( a_{ij} \) from any row or column of \( A \).
2. Form the submatrix: Remove the i^th row and j^th column from \( A \) to obtain the submatrix \( A_{ij} \).
3. Calculate the minor: The minor \( M_{ij} \) is the determinant of the submatrix: \[ M_{ij} = |A_{ij}| \]
4. 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.
5. 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)