# 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.

• 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
• LetΥs 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.

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