1. Consider problem of finding n real numbers x1, x2, . . . , xn which satisfy m equations

a11x1 + a12x2 + · · · + a1nxn = y1

a21x1 + a22x2 + · · · + a2nxn = y2

am1x1 + am2x2 + · · · + amnxn = ym

where y1, y2, . . . , ym and aij (1 ≤ i ≤ m, 1 ≤ j ≤ n) are given real numbers. In general, m ≥ n. The above is sometimes written as x1a1 + x2a2 + . . . + xnan = y. where ai = [a1i, a2i. . . , ami]T and yi = [y1, y2 . . . , ym]T, or as Ax = y where

A =

a11 a12 · · · a1n

a21 a22 · · · a2n

am1 am2 · · · amn



x = [x1, x2, . . . , xn]T and yi = [y1, y2 . . . , ym]T. Suppose m = n, and the matrix A = [aij ] is invertible. Describe method of finding a solution x = [x1, x2, . . . , xn]T by inverting A using method of determinants. What would be time complexity?

2. Two systems of linear equations are called equivalent if each equation in one is a linear combination of the equations in the other. Argue that equivalent systems of linear equations have exactly the same solutions.

3. Show that to each elementary row operation e there corresponds an elementary row operation e1, of the same type as e, such that e1(e(A)) = e(e1(A)) = A for each A. In other words, every elementary row operation has an inverse which is an elementary row operation of the same type.

4. Consider the following systems of equation

(a)

x1 − x2 + 2×3 = 1

2×1 + 2×3 = 1

x1 − 3×2 + 4×3 = 2

(b)

x1 − 2×2 + x3 + 2×4 = 1

x1 + x2 − x3 + x4 = 2

x1 + 7×2 − 5×3 − x4 = 3

Find out whether they have solutions. If so, explicitly describe all solutions.