We have an inconsistent system given by Ax = b with matrix A is a 3x3 matrix.
Which of the following is true? (a) rref(A) has at least one row full of zeroes (b) im(A) is not R^3 (c) ker(A) does not equal to 0
I can now conclude that rref(A) has at least one row full of zeroes by identification of inconsistent system.
Then by extension, b is true since the column space will be missing one leading one.
I'm uncertain about part C
The fact that $Ax= b$ does not have a solution, means that $b$ does not lie in the range of $A$. Consequently, $\mbox{im } A$ cannot equal $\mathbb R^3$ as it does not contain $b$.
Since $A$ is not a surjective map, then by the rank-nullity theorem, $A$ is not an injective map, because $A : \mathbb R^3 \to \mbox{im } A$ mut satisfy $\dim \ker A + \dim \mbox{im } A = 3$, but $\dim \mbox{im} A < 3$ so $\dim \ker A > 1$, which means that the kernel is non-trivial.
Finally, the row rank of $A$, equals the column rank of $A$, equals the dimension of the image of $A$, which is not equal to $3$. Since there are three rows, and the image space has smaller dimension, row reduction will lead to a row of zeroes.
So the answer to all three is yes.