The purpose of determining A−1 using elementary row operations is to find the inverse of a given matrix A. The inverse of a matrix is useful in solving systems of linear equations and in other mathematical operations.
Elementary row operations are a set of three operations that can be performed on a matrix: 1) multiplying a row by a non-zero constant, 2) adding a multiple of one row to another row, and 3) interchanging two rows. These operations are used to manipulate a matrix in order to find its inverse.
To determine A−1 using elementary row operations, you need to perform the same operations on both the given matrix A and the identity matrix of the same size. The identity matrix is a square matrix with 1's on the main diagonal and 0's everywhere else. Once the given matrix is transformed into the identity matrix, the resulting matrix will be the inverse of A.
The steps involved in determining A−1 using elementary row operations are:
No, A−1 can only be determined for square matrices that are invertible. A square matrix is invertible if its determinant is non-zero. If the determinant of a square matrix is zero, it is called singular and its inverse does not exist.