Adjoint of a matrix, Inverse of a matrix

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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 = B^T.

Theorems.

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

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

4. If A and B are square matrices adj AB = adj B · adj A

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, ... , H_n are the elementary matrices, then A^-1 = H_n, ... ,H₂, H₁.

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

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

Inverse of a direct sum. If A₁, A₂, ... ,A_s are nonsingular square matrices, then the inverse of the direct sum diag(A₁, A₂, ... ,A_s ) is diag(A₁^-1, A₂^-1, ... ,A_s^-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, .... ,H_s be the elementary matrices corresponding to those elementary row operations. Then

H_s.... H₂H₁A = I

Since A^-1A = I it follows that

A^-¹ = H_s.... 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^-1is equal to the product in reverse order of the corresponding elementary matrices.

References.

Ayres. Matrices (Schaum).