Question

1. Choose one problem, mark it, and solve it. (10 points) A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Justify your answer. (8 points) If a 6 x 4 matrix A has rank 2, find dim NulĄ and dim RowA.

Answer 1

a) The nehomogeneous system of ten linear ears in tuelve unknowns. It has three free variable Hente, tant Tet the mateix equation is AX =B A is 10x12 where matix I B is 1001 matrix Here f(A) = 9 < no. of rows of A. flow, If the Eight_side of the recitation Equation changes te B changes to B' Then the equation is Ax=B' It S[AIBJ = 9 then the equation have a solution But if S[AIB] = 10, then S[A] # SAIB] Thus, the system is in consistant. Hence, the non homogeneous equation will not have a stouti Solution always. A is 6x4 matrix & S(A)=2 # 1 (A) = dim (Row A) = dim (Row A) =? By Eank-nuility thm. dim (u) = S(A) + nullity (Abof a Here dim (u) = 4 L = 2+ na NAL =) | dim (NukА)) = 2