Solution space of a linear system

Question

Whats an example of a 2x3 matrix $A$, if possible such that the solution space of the linear system $Ax=0$ is $\mathbb R^3$? I know the zero matrix is one, but does there exist a non-zero matrix?

Answer 1

Suppose that $$ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}$$

is a real $2\times 3$ matrix. If $A\vec{x}=\vec{0}$, where $$\vec{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix},$$

then we must have $$a_{11}x_1 + a_{12}x_2 + a_{13}x_3=0 \mbox{ and } a_{21}x_1 + a_{22}x_2 + a_{23}x_3=0.$$ If $A$ is not the zero matrix, then the solution set will be the intersection of two planes (or the intersection of a plane and $\mathbb{R^3}$ if one of the rows of $A$ is filled with zeros). Clearly, this cannot be the whole of $\mathbb{R}^3$, and will in fact be either a plane or a line.

---

If you would allow me to speak slightly more abstractly, I may be able to better answer your question. First, let's go over some terminology. Any terms which you feel need to be explained further, just look them up on Wikipedia.

• The kernel of an $m \times n$ matrix $A$ is the set $\ker(A)=\{\vec{x}\in\mathbb{R}^n : A\vec{x}=\vec{0}\}$. This is just the solution set of $A\vec{x}=\vec{0}$.
• The nullity of $A$ is then the dimensions of the kernel of A.

Here is a very important fact:

The only $m \times n$ matrix with nullity $n$ is the zero matrix of that size.

Suppose that $A$ is an $m\times n$ real matrix. The fact demonstrates that $$\mbox{ the solution space of } A\vec{x}=\vec{0} \mbox{ is } \mathbb{R}^n \implies A \mbox{ is the zero matrix. }$$ Clearly, we have the converse. So, $$\mbox{ the solution space of } A\vec{x}=\vec{0} \mbox{ is } \mathbb{R}^n \iff A \mbox{ is the zero matrix. }$$

Answer 2

The answer is no. Because of rank nullity theorem any such matrix $A$ must have rank zero. And only matrix with rank zero is the zero matrix.