Adjoint of a Square Matrix

Adjoint of a Square Matrix:

The adjoint of a square matrix A, denoted as adj(A) or A, is a matrix formed by taking the transpose of the matrix of cofactors (also known as the cofactor matrix) of A.

Calculation:

For a square matrix A of size n×n:

  1. Cofactor Matrix: Calculate the matrix of cofactors C for matrix A. Each element cij of C is the determinant of the submatrix obtained by removing the ith row and jth column of A, multiplied by (1)i+j.
  2. Transpose: Transpose the matrix C to obtain the adjoint adj(A) or A.

Example:

Consider a matrix A:

A=[2354]
  1. Cofactor Matrix:
C=[4532]
  1. Transpose:
adj(A)=CT=[4352]

Properties of the Adjoint:

  1. Inverses: For a non-singular matrix A, A1=1det(A)adj(A), where det(A) is the determinant of A.

  2. Non-Singular Matrix: If det(A)0, A is non-singular, and its adjoint exists.

  3. Product with Matrix: For any matrix B, A(adj(A)B)=(adj(A)BA)=(det(A)I)B=det(A)B