Determinants

In the previous chapter, we have studied inverse of a matrix. In this section, we shall discuss the condition for existence of inverse of a matrix. To find inverse of a matrix, we shall first define adjoint of a matrix.

4.5.1 Adjoint of a matrix

Definition 3 The adjoint of a square matrix \(A = [a_{ij}]_{n \times n}\) is defined as the transpose of the matrix \([A_{ij}]_{n \times n}\), where \(A_{ij}\) is the cofactor of the element \(a_{ij}\). Adjoint of the matrix A is denoted by adj A. Let \(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\) Then \(adj A = Transpose of \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} = \begin{bmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{bmatrix}\)