Find the null space of the following matrix.

Question

Let $M_{m\times n}(\mathbb{R})$ be the set of all $m\times n $ matrices with real entries. Which of the following matrices are correct.

1. There exists a $A\in M_{2\times 5}(\mathbb{R})$ such that the dimension of the null space of $A$ is 2.
2. There exists a $A\in M_{2\times 5}(\mathbb{R})$ such that the dimension of the null space of $A$ is 0.
3. There exists a matrix $B\in M_{5\times 2}$ such that $AB$ is the identity matrix.
4. There exists a $A\in M_{2\times 5}(\mathbb{R})$ whose null space is $\{(x_1,x_2,x_3,x_4,x_5)\in \mathbb{R^5}: x_1=x_2,x_3=x_4=x_5\}$

Since $\text{rank}A\leq$min$(m,n)$ so rank $A\leq2$. By rank nullity theorem we can conclude that $\text{nullity}A\geq3$. Hence option 1 and 2 are false. Option 3 is true, easy to check. But I am confused with the option 4. I don't know how to reason with that. Can anybody help me with any ideas or hints? Thanks.

Answer 1

It's the number of free variables you have to count, that's the dimension of the nullspace.

Answer 2

Only 3 is true as dimensions of null space should be atleast 3. In last option dimension of null space is 2.