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Inverting matrices using simple Gaussian elimination

The task of finding the inverse matrix $\hspace{0.2ex}\underline{\underline{X}}{}\hspace{0.15ex}$ of the square Nth order matrix $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}$ can be stated in the following way

$$\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex} \h... ...ex} = \hspace{0.2ex}\underline{\underline{I}}{}\hspace{0.15ex}$$ (2.15)

where the matrix $\hspace{0.2ex}\underline{\underline{I}}{}\hspace{0.15ex}$ is the identity matrix of order N and we assume that $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}$ has full rank. The matrix $\hspace{0.2ex}\underline{\underline{X}}{}\hspace{0.15ex}$ is most commonly found in a column wise manner since we may rewrite (2.22) in the way

$$\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex} \h... ...rline{I}{}\hspace{0.15ex}_i, \hspace{1.5cm} i=\{1,2,\ldots,N\}$$ (2.16)

where $\hspace{0.2ex}\underline{X}{}\hspace{0.15ex}_i$ and $\hspace{0.2ex}\underline{I}{}\hspace{0.15ex}_i$ are the ith column of the matrices $\hspace{0.2ex}\underline{\underline{X}}{}\hspace{0.15ex}$ and $\hspace{0.2ex}\underline{\underline{I}}{}\hspace{0.15ex}$ respectively. We realize that we have to solve a Nth order system of equations N times with the same coefficient matrix and different right hand side vectors. This can be accomplished by simple Gaussian elimination, ie Gaussian elimination without pivoting. It is important to emphasize that simple Gaussian elimination does not take into account the limited machine accuracy when doing the calculations in practice. In unfavourable circumstances the solution can be totally destroyed by rounding errors. These examples are, however, very seldom met in practice and we assume simple Gaussian elimination to suffice for our purposes. For completeness we include the algorithm [9, p. 2.14] of simple Gaussian elimination below. In the algorithm A is the coefficient matrix and B the right hand side vector.

$$\begin{eqnarray*} \parbox[t]{4.5cm}{1) Forward elimination:}\parbox[t]{15cm}{ ... ...,i)}$ }\\ \parbox[t]{4.5cm}{}\parbox[t]{15cm}{ {\bf next} i } \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

This algorithm overwrites the right hand side vector, B, with the solution vector. When we wish to solve for different right hand side vectors we only need to perform the first part down to the comment statement { Manipulate right hand side} once.

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