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Linear Algebra

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All Variables Free

LINALG-JHNLHB

Suppose that you are given a linear system $A \mathbf{x} = \mathbf{b}$ and you are told that (1) the system is consistent and (2) every variable is a free variable.

Which of the following is TRUE?

A

$A$ is the zero matrix and $\mathbf{b}$ is a vector of all $0$'s.

B

$A$ is the identity matrix and $\mathbf{b}$ is a vector of all $0$'s.

C

The matrix $A$ has rank equal to the number of its columns.

D

The augmented matrix $[A | \mathbf{b} ]$ has rank equal to the number of its columns.