It follows that A is a square matrix and both A -1 and A has the same size.
A matrix is said to be invertible, non-singular, or non–degenerative if it satisfies this condition. So, if we consider B = A -1, then AA -1 = A -1 A = Iįor a matrix to be invertible, the necessary and sufficient condition is that the determinant of A is not zero. Therefore, by definition, if AB = BA = I, then B is the inverse matrix of A and A is the inverse matrix of B. Inverse of a matrix is defined as a matrix which gives the identity matrix when multiplied together. Therefore, the proper term is adjugate matrix or adjunct matrix. The term “adjoint” is rather outdated and now used for complex conjugate of a matrix. The adjoint can be used to compute the Inverse of a matrix and for finding the derivative of a determinant by the Jacobi’s formula. The determinant of the matrix obtained by removing the i th row and j th column is known as the minor of the ij th element. i.e adj( A) = C T.Ĭofactor matrix is given by C = (-1) i+j M ij, where M ij is the minor of the ij th element. If the cofactor matrix of A is C, then the adjugate matrix of A is given by C T. The adjoint matrix, or the adjugate matrix is the transpose of the cofactor matrix. More about (Classical) Adjoint or Adjugate Matrix Both adjoint matrix and the inverse matrix are obtained from linear operations on a matrix, and they are two different matrices with different properties.
