Adjoint Matrix:

Definition: The transpose of the matrix obtained by replacing the elements of a square matrix A by the corresponding cofactors is called the adjoint matrix of A. It is denoted by Adj A or adj A.

Adjoint and Inverse of a Matrix:

The adjoint of a square matrix [aij] is defined as the transpose of the matrix [Aij] where Aij are the cofactors of the elements aij. Adjoint of A is denoted by adj A.
Let A be a square matrix of order n. If there exists a matrix B of order n such that AB = BA = I, where I is the identity matrix of order n, then the matrix A is said to be invertible and B is called the inverse (or reciprocal) of A.
A square matrix A is said to be non-singular if its determinant value is non-zero.
A square matrix A is said to be singular if |A| = 0.