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Adjoint of a matrix, Inverse of a matrix

Adjoint of a square matrix. The transpose of the matrix obtained by replacing each element by its cofactor. If B is the matrix obtained by replacing each element of a square matrix A by its cofactor, then adj A = BT .

Theorems.

1. A(adj A) = (adj A)A = diag ( |A|, |A|, ..., |A| ) = |A| · In

2. If A is n-square and nonsingular, | adj A | = |A| n-1

3. If A is n-square and singular, A(adj A) = (adj A)A = 0

Inverse of a matrix. For a nonsingular square matrix, the inverse is the quotient of the adjoint of the matrix and the determinant of the matrix. i.e .the inverse A-1 of a matrix A is given by

The inverse is defined only for nonsingular square matrices.

The following relationship holds between a matrix and its inverse:

AA-1 = A-1A = I

where I is the identity matrix.

Theorems.

1] A square matrix has an inverse if and only if it is nonsingular.

2] The inverse of a nonsingular square matrix is unique.

3] For matrices A, B and C, if A is nonsingular, then AB = AC implies B = C.

4] A nonsingular square matrix can be reduced to normal form by row transformations alone.

5] If a nonsingular matrix A is reduced to the identity matrix I by a sequence of row transformations alone, then A-1 is equal to the product in reverse order of the corresponding elementary matrices i.e. if H1, H2, ... , Hn are the elementary matrices, then A-1 = Hn, ... ,H2, H1 .

Inverse of a nonsingular diagonal matrix. The inverse of the nonsingular diagonal matrix

diag (k1,k2, ... ,kn) is the diagonal matrix diag (1/k1, 1/k2, ..., 1/kn ) .

Inverse of a direct sum. If A1, A2, ... ,As are nonsingular square matrices, then the inverse of the direct sum diag(A1, A2, ... ,As ) is diag(A1-1, A2-1, ... ,As-1 ) .

Computing the inverse of a non-singular matrix. Let a nonsingular matrix be reduced to normal form by elementary row operations and let H1, H2, .... ,Hs be the elementary matrices corresponding to those elementary row operations. Then

Hs .... H2H1A = I

Since A-1A = I it follows that

A-1 = Hs .... H2H1

Using this fact we can compute the inverse of a nonsingular matrix A by adjoining the identity matrix to the right of it and then reducing A to an identity matrix. As we reduce A to an identity matrix, the identity matrix adjoined to the right of it will be transformed into A-1.

Example. Find the inverse of

.

Thus as A is reduced to I, I is carried into

.

Theorem. If a matrix A is reduced to I by a sequence of row transformations alone, then A-1 is equal to the product in reverse order of the corresponding elementary matrices.

References.

Ayres. Matrices (Schaum).