Given a nonsingular system of linear equations Axb what effe

Given a nonsingular system of linear equations Ax=b, what effect on the solution vector x results from each of the following actions? Permuting the rows of [A b] Permuting the columns of A Multiplying both sides of the equation from the left by a nonsingular matrix M

Solution

Given non-singular system of linear equations is

Ax = b

(a) If we\'ll permute the rows of [ A b ], then there will be no effect on the solution vector because we can use same elementary row operations on both sides of the above equation.

(b) But since, same column operations we can use on both sides of above equation because there is only one column in the vector b. Hence solution vector will not be correct after permuting the columns of A.

(c) If we multiply a non singular matrix M from the left of both sides, then there will be no effect on the solution vector because by the properties of matrix we can pre or post multiply any nonsingular matrix on both sides of a matrix equation. (matrix M should be of a proper order so that the product matrix on both sides should have same order.)

For verification of these answers we can take any example as any matrix A of 2x2 order, x of order 2x1 and b of order 2x1.