Question

7. Suppose A is a 6 x 3 matrix with 3 pivot positions. (a) Does the equation Ax O have a nontrivial solution? (b) Does the equation Ax =b have at least one solution for every b E R6? %3D

Answer 1

7. (a). If A is a 6 x 3 matrix with 3 pivot positions, it means that the 3 columns of A are linearly independent. Further, since A is not a square matrix, the equation AX = 0 does not have a unique solution. Thus, the equation AX = 0 will have infinite solutions ( a homogeneous equation AX = 0 is always consistent). Therefore, there will be non-trivial solutions to the equation AX = 0.

(b). Since A is a 6 x 3 matrix with 3 pivot positions,, hence every b in R³ need not be a linear combination of the columns of A. Therefore, the equation AX = b need not have a solution for every b in R⁶.