Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 82

Matrix inversion has another serious drawback—a bandedmatrix loses its struc-
ture during inversion. In otherwords, if A is banded or otherwise sparse, then A − 1 is
fullypopulated. However, the inverse of a triangular matrix remainstriangular.

EXAMPLE 2.13
Write a function that inverts amatrix using LU decompositionwith pivoting. Test the
function by inverting

⎡
⎣

⎤
⎦

0 . 6 − 0 . 4 1 . 0

A = − 0 . 3 0 . 2 0 . 5

0 . 6 − 1 . 00 . 5

Solution The function matInv listedbelow inverts any martix A .

function Ainv = matInv(A)
% Inverts martix A with LU decomposition.
% USAGE: Ainv = matInv(A)

n = size(A,1);
Ainv = eye(n); % Store RHS vectors in Ainv.
[A,perm] = LUdecPiv(A); % Decompose A.
% Solve for each RHS vector and store results in Ainv
% replacing the corresponding RHS vector.
fori=1:n
Ainv(:,i) = LUsolPiv(A,Ainv(:,i),perm);

end

The following test program computes the inverse of the givenmatrix and checks
whether AA − 1 = I :

% Example 2.13 (Matrix inversion)
A=[0.6-0.41.0
-0.3 0.2 0.5
0.6 -1.0 0.5];
Ainv = matInv(A)
check = A*Ainv

Here are the results:

>> Ainv =
1.6667 -2.2222 -1.1111

1.2500 -0.8333 -1.6667

0.5000 1.0000 0