Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 20/Solution 1

The set of solutions to the homogeneous system has $2$ parameters, so the dimension of the null space of this matrix is $2$.

The dimension of the domain is $4$. By the Rank-Nullity Theorem, we have $\mathrm {rank} (A)+\mathrm {null} (A)=4$.

Thus, the rank of $A$ is $\color {blue}2$.