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
  • Let՚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.

Properties of Determinants

Illustration 2 for Properties_of_Determinants

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