MATRIX ALGEBRA:DETERMINANTS,INVERSES,EIGENVALUES

§C.3 MATRIX INVERSION

The inverse of a square nonsingular matrix A is re presented by the symbol A⁻¹ and is defined by the
relation

The most important application of the concept of inverse is the solution of linear systems. Suppose
that, in the usual notation, we have

Ax = y (C.32)

Premultiplying both sides by A⁻¹ we get the inverse relationship

More generally, consider the matrix equation for multiple (m) right-hand sides:

which reduces to (C.32) for m = 1. The inverse relation that gives X as function of Y is

In particular, the solution of

is X = A⁻¹. Practical methods for computing inverses are based on directly solving this equation ; see
Remark C.4.

§C.3.1 Explicit Computation of Inverses

The explicit calculation of matrix inverses is seldom needed in large matrix computations. But ocassionally
the need arises for the explicit inverse of small matrices that appear in element computations.
For example, the inversion of Jacobian matrices at Gauss points, or of constitutive matrices.
A general formula for elements of the inverse can be obtained by specializing Cramer’s rule to (C.36).
Let B =Then

in which denotes the so-called adjoint of entry ofA. The adjoint is defined as the de terminant
of the submatrix of order (n − 1) × (n − 1) obtained by deleting the j^th row and i^th column of A,
multiplied by

This direct inversion procedure is useful only for small matrix orders: 2 or 3. In the examples below
the inversion formulas for second and third order matrices are listed.

EXAMPLE C.9
For order n = 2:

in which |A| is given by (C.2).

EXAMPLE C.10
For order n = 3:

where

in which |A| is given by (C.3).

EXAMPLE C.11

If the order exceeds 3, the general inversion formula based on Cramer’s rule becomes rapidly useless
as it displays combinatorial complexity . For numerical work it is preferable to solve the system (C.38)
after A is factored. Those techniques are described in detail in linear algebra books ; see also Remark
C.4.

§C.3.2 Some Properties of the Inverse

I. The inverse of the transpose is equal to the transpose of the inverse:

because

II. The inverse of a symmetric matrix is also symmetric. Because of the previous rule,
hence is also symmetric.

III. The inverse of a matrix product is the reverse product of the inverses of the factors:

This is easily verified by substituting both sides of (C.39) into (C.31). This property generalizes
to an arbitrary number of factors.

IV. For a diagonal matrix D in which all diagonal entries are non zero , D⁻¹ is again a diagonal matrix
with entries 1/d_ii . The verification is straightforward

V. If S is a block diagonal matrix:

then the inverse matrix is also block diagonal and is given by

VI. The inverse of an upper triangular matrix is also an upper triangular matrix. The inverse of a
lower triangular matrix is also a lower triangular matrix. Both inverses can be computed in O(n²)
floating-point ope rations .

REMARK C.4

The practical numerical calculation of inverses is based on triangular factorization. Given a nonsingular n × n
matrix A, calculate its LU factorization A = LU, which can be obtained in O(n³) operations. Then solve the
linear triangular systems:

UY = I, LX = Y, (C.47)

and the computed inverse A⁻¹ appears in X. One can overwrite I with Y and Y with X. The whole process can be
completed in O(n³) floating-point operations. For symmetric matrices the alternative decomposition A = LDL^T ,
where L is unit lower triangular and D is diagonal, is generally preferred to save computing time and storage.