# Adjoint and Inverse of a Matrix: Adjoint of a Matrix: Relation between Adjoint and Inverse of a Matrix (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Title: Adjoint and Inverse of a Matrix

- The adjoint of a matrix (also called the adjugate of a matrix) is defined as the transpose of the cofactor matrix of that matrix.
- The Inverse of a Matrix exists if and only if the value of its determinant is equal to zero
- For a matrix A, the adjoint is denoted as
**adj (A)**. - While the inverse of a matrix A is that matrix which when multiplied by the matrix a give an identity matrix. The inverse of a Matrix A is denoted by
**A**.^{-1} - The adjoint of a square matrix A = [aij] n x n is defined as the transpose of the matrix [Aij] n x n, where Aij is the cofactor of the element aij. Adjoing of the matrix A is denoted by
**adj A**.

## Adjoint of a Matrix

- Thus if
- Adj A = Transpose of
- of

## Relation between Adjoint and Inverse of a Matrix

To find the inverse of a matrix A, i.e.. , we shall first define the adjoint of a matrix. Let A be an n x n matrix. The (i, j) cofactor of A is defined to be

- Where, is the minor matrix obtained from A after removing the ith row and column
- LetΥs consider the matrix and define the matrix
- The matrix is called the adjoint of matrix A.
- When A is invertible, then its inverse can be obtained by the formula given below.