An Efficient and Simple Algorithm for Matrix Inversion

An Efficient and Simple Algorithm for Matrix Inversion

Ahmad Farooq (King Khalid University, Saudi Arabia) and Khan Hamid (National University of Computer and Emerging Sciences (NUCES), Pakistan)
DOI: 10.4018/978-1-4666-1752-0.ch002
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In this paper, a new algorithm is proposed for finding inverse and determinant of a given matrix in one instance. The algorithm is straightforward in understanding and manual calculations. Computer implementation of the algorithm is extremely simple and is quite efficient in time and memory utilization. The algorithm is supported by an example. The number of multiplication/division performed by the algorithm is exactly; however, its efficiency lies in the simplicity of coding and minimal utilization of memory. Simple applicability and reduced execution time of the method is validated form the numerical experiments performed on test problems. The algorithm is applicable in the cases of pseudo inverses for non-square matrices and solution of system of linear equations with minor modification.
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(A). Simple Algorithm For Matrix Inversion

The algorithm assumes to take a square matrix of dimension n. The inverse is calculated in n iterations. In each iteration p, all the existing elements of A change to new values After the last iteration i.e. when , will be the elements of the inverse. The determinant of the matrix (denoted by d) is also calculated iteratively through successive multiplication of the pivot selected in each iteration. In this algorithm the pivots are selected diagonally starting from to If any pivot is found to be zero i.e., then inverse cannot be calculated. If an inverse is calculated then d will contain the determinant of A.

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