Suppose that $[A|b]$ is the augmented matrix of a system of 7 linear equations in 10 variables.

Question

Suppose that $[A|b]$ is the augmented matrix of a system of 7 linear equations in 10 variables. For the following cases, determine which of the following statements (a) system is inconsistent (b) consistent (c) not enough information to determine (d) data are impossible

(1) rank(A)=7=rank(A|b).

(2) rank(A)=5, rank(A|b)=6

(3) rank(A)=6, rank(A|b)=5

(4) rank(A)=7, rank(A|b)=8

(5)rank(A)=5, rank(A|b)=7

I am not sure my following answer is right.

For (1), I think this should be (b). Since if rank(A|b)=rank(A), then the system of linear equations is consistent. But how many parameters are involved in the general solution?

For (2). This system is (a).

For (3). This system is (d)

For (4), this system is (a)

For (5), this is (d).

Answer 1

For 1: that's right, and the number of independent free variables is the dimension of the nullspace which is $n-r=10-7=3$.

In 2-5 you have one wrong answer. Hint: pay attention to the shapes of $A$ and $A|b$.