1. What Is the Adjoint of a Matrix?
The adjoint (or adjugate) of a square matrix is the matrix obtained by taking the transpose of its cofactor matrix. It plays an important role in finding the inverse of a matrix.
For a matrix \( A \), the adjoint is written as:
\text{adj}(A)
2. Definition of Adjoint
If \( A = [a_{ij}] \) is a square matrix, then:
- Find the cofactor of each element \( a_{ij} \).
- Arrange these cofactors in a matrix (cofactor matrix).
- Take the transpose of this cofactor matrix.
The resulting matrix is the adjoint of \( A \).
3. Steps to Find adj(A)
- Find the minor of each element of the matrix.
- Compute the cofactor using \( C_{ij} = (-1)^{i+j} M_{ij} \).
- Write the cofactor matrix.
- Take the transpose of the cofactor matrix.
3.1. Example: Finding adj(A)
Let:
A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}
Step 1: Find minors:
- Minor of 2 = 4
- Minor of 3 = 1
- Minor of 1 = 3
- Minor of 4 = 2
3.2. Step 2: Compute Cofactors
Using \( C_{ij} = (-1)^{i+j} M_{ij} \), we get:
\begin{bmatrix} +4 & -1 \\ -3 & +2 \end{bmatrix}
3.3. Step 3: Take Transpose
Transpose of the cofactor matrix gives:
\text{adj}(A) = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}
4. Using Adjoint to Find the Inverse
The adjoint is used in the formula for the inverse of a matrix:
A^{-1} = \frac{1}{\det(A)} \text{adj}(A)
This formula works only when the determinant is non-zero.
5. Properties of adj(A)
- \( A \cdot \text{adj}(A) = \det(A) \, I \)
- \( \text{adj}(I) = I \)
- \( \text{adj}(AB) = \text{adj}(B) \cdot \text{adj}(A) \) (order reverses)
- \( \text{adj}(kA) = k^{n-1} \text{adj}(A) \) for an \( n \times n \) matrix
6. Why Adjoint Is Important
The adjoint is used for:
- finding the inverse of a matrix
- proving identities involving matrices
- theoretical results in linear algebra
It serves as a bridge between determinants and inverses.