 Solution

For $p=1$, the third coefficient row equals three times the second minus twice the first. The first two rows are independent, so the coefficient matrix rank is two. The same dependence must hold for the right-hand side, givingThus infinitely many solutions occur exactly for $p=1$ and $t\in \{1,2\}$. The null space has dimension one by the rank-nullity theorem, and an explicit parameterization, with $s\in \mathbb{R}$, isSubstitution verifies all three equations in each family.