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Solvability of Linear Systems of Equations and Matrix Inversion with MATLAB

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A linear system $ Ax=b$ can be solved in MATLAB with the command x=A\b.
Depending on the dimensions different techniques are used.

In case of numerically instable matrices or in case of systems with loss of rank a warning will be displayed. (The matrix is close to singular or badly scaled.)

The MATLAB function inv inverts a regular square matrix:

$\displaystyle A \_ inverse = {\tt inv(A)}. $

Hence, an $ n \times n$ linear system can also be solved via

$\displaystyle {\tt x=inv(A)*B}. $

However, calculating the solution by x=A\b should definitely be preferred, since the \-command uses more efficient and more stable algorithms.

Finally,

$\displaystyle [Q, R, P] = qr(A) $

generates the QR-factorization of an $ m \times n$-matrix A:

$\displaystyle A P = Q R, $

where $ P$ is a permutation matrix, $ Q$ is unitary and $ R$ generalized upper triangular.

see also:

[Examples]
  automatically generated 3/ 8/2007