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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 = 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^{-1}A = 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 H_{1}, H_{2}, ... , H_{n} are the elementary matrices, then A^{-1} = H_{n}, ... ,H_{2}, H_{1 }.

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

diag (k_{1},k_{2}, ... ,k_{n}) is the diagonal matrix diag (1/k_{1}, 1/k_{2}, ..., 1/k_{n} ) .

Inverse of a direct sum. If A_{1}, A_{2}, ... ,A_{s} are nonsingular square matrices, then the inverse of the
direct sum diag(A_{1}, A_{2}, ... ,A_{s} ) is diag(A_{1}^{-1}, A_{2}^{-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 *H _{1}, H_{2}, .... ,H_{s} * be the elementary
matrices corresponding to those elementary row operations. Then

*H _{s }.... H_{2}H_{1}A = I*

Since A^{-1}A = I it follows that

*A ^{-}^{1} = H_{s }.... H_{2}H_{1}*

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).

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