Adjoint vs Inverse 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.
More about (Classical) Adjoint or Adjugate Matrix
The adjoint matrix, or the adjugate matrix is the transpose of the cofactor matrix. If the cofactor matrix of A is C, then the adjugate matrix of A is given by CT. i.e adj(A) = CT.
Cofactor matrix is given by C = (-1)i+j Mij, where Mij is the minor of the ijth element. The determinant of the matrix obtained by removing the ith row and jth column is known as the minor of the ijth element. [To compute the adjugate matrix, first find the minors of each element, then form the cofactor matrix, finally taking the transpose of that gives the adjugate 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 The term “adjoint” is rather outdated and now used for complex conjugate of a matrix. Therefore, the proper term is adjugate matrix or adjunct matrix.
More about Inverse Matrix
Inverse of a matrix is defined as a matrix which gives the identity matrix when multiplied together. Therefore, by definition, if AB = BA = I, then B is the inverse matrix of A and A is the inverse matrix of B. So, if we consider B = A-1, then AA-1 = A-1A = I
For a matrix to be invertible, the necessary and sufficient condition is that the determinant of A is not zero. i.e |A| = det(A) ≠ 0. A matrix is said to be invertible, non-singular, or non–degenerative if it satisfies this condition. It follows that A is a square matrix and both A-1 and A has the same size.
The inverse of the matrix A can be calculated by many methods in linear algebra such as Gaussian elimination, Eigendecomposition, Cholesky decomposition and Carmer’s rule. A matrix can also be inverted by block inversion method and Neumann series.
The Cramer’s rule provides an analytical method of finding the inverse of a matrix, and the non-singularity condition can also be explained by the results. By Cramer’s rule A-1 = adj(A)/det(A) or adj(A) = A-1 det(A). For this result to be valid, det(A) ≠ 0, hence matrices are invertible if and only if the above condition is satisfied.
What is the difference between Adjoint and Inverse Matrices?
• The adjugate or adjoint of a matrix is the transpose of the cofactor matrix, whereas inverse matrix is a matrix which gives the identity matrix when multiplied together.
• Adjugate matrix can be used to calculate the inverse matrix and is one of the common methods of finding the inverses manually.
• For every matrix, an adjugate matrix exists, but the inverse exists if and only if the determinant is non-zero.