Matrixinversionproblem矩阵的求逆问题.doc

Matrix inversion problem. Given a non-singular square matrix , obtain a matrix such that . The matrix is the inverse of . In our course, we would be considering only inverses of non-singular matrices over real field (no complex matrix).Matrix inversion through its adjoint.Good for small matrices. Consider the matrix . Let the corresponding cofactor matrix be where , etc. Then the inverse of is This scheme is impractical for large matrices. Consequently, we need easier approaches to deal with the problem. We would deal with the square matrices only. Weassume its determinant is non-zero.Given a non-singular square matrix we can obtain its inverse . We will approach this problem from different angles. a. Using elementary matrices: A matrix is an elementary matrix if it is obtained from an identity matrix by a single row operation. e.g We can generate a number of elementary matrices from it. a. b. c. d. A general matrix can be expressed as a single column 3 rows: Let What is the effect of operating by an elementary matrix on A? For instance, , interchange. is replaced by , by Replace by and These demonstrate the effect of elementary matrices on general matrices - they effectively achieve row-operations. Therefore, using such matrices, we can transform a non-singular matrix A into its row-echelon form. Thus, Therefore, the matrix product This gives us a procedure to obtain inverse of a non-singular matrix A using row transformation.a. Start with an augmented matrix . b. Carry out row-transformation on this using elementary matrices.c. When the left-side becomes an identity matrix, the transformed right side must be the inverse of the original matrix . Observe:a. Two matrices and are row equivalent to each other if one can get from using a sequence of elementary matrices on the latter. That means b. Every elementary row-operation can be “undone” by another elementary row-operation. Therefore, every elementary matrix has an inverse.c. The inverse of a product is the product of the inverses in reverse order. For instance,d. Finally, given any the following statements are equivalent:1. has an inverse.2.