The adjugate of a matrix is also known as the adjoint or classical adjoint. It is denoted by adj(A).

  To find the adjugate of a matrix A, we need to find the cofactor matrix of A and then take its transpose.

  The cofactor of each element of matrix A is the determinant of the submatrix obtained by removing the row and column containing that element, multiplied by (-1) raised to the power of the sum of the row number and column number. Mathematically, the cofactor Cij of the element Aij is given by:

  Cij = (-1)^(i+j) * det(Mij)

  where Mij is the submatrix obtained by removing the i-th row and j-th column from A.

  After obtaining the cofactor matrix C, we take its transpose to get the adjugate matrix adj(A).

  In summary, the steps to find the adjugate of matrix A are:

  1. Compute the cofactor of each element of A using the formula mentioned above.

  2. Construct the cofactor matrix C using the computed cofactors.

  3. Take the transpose of C to get the adjugate matrix adj(A).

  The adjugate of a matrix has the property that when multiplied by the original matrix, it gives the scalar multiple of the identity matrix, i.e., A * adj(A) = det(A) * I. This property is essential in finding the inverse of a matrix.

  The size of the adjugate matrix is the same as the size of the original matrix, i.e., if A is an n x m matrix, then adj(A) is also an n x m matrix.