** 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 *C*^{T}. i.e adj(*A*) = *C*^{T}.

Cofactor matrix is given by *C *= (-1)^{i+j} *M*_{ij}, where *M*_{ij} is the minor of the ij^{th} element. 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. [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 The term “adjoint” is rather outdated and now used for complex conjugate of a matrix.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*^{-1}*A *= *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.

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