Prove or provide counterexample: if the coefficient matrix of a system of

linear equations in

unknowns has rank

, then the system has a solution.

I have an idea for the proof but I am not sure how to formalize it. Some advice, please?

Here's the proof: let

be the coefficient matrix in the system

. If

has rank of

, then

. Say that's not true - then

, implying that the maximum rank of

which equivalent to the max number of linearly independent columns, has a maximum of

, which is less than

by assumption, meaning that the rank will never equal

. So we must have

. Then if the rank of

is

, we can reduce the matrix [A|b] using row operations to a form [A'|b'] where all

have coefficients of

. Then a solution becomes

for

and for

, we have

.