Question-87

Two non-zero square matrices \(\displaystyle A\) and \(\displaystyle B\) of order \(\displaystyle n\) are given to you with the property that \(\displaystyle Ax=0\) and \(\displaystyle Bx=0\) have the same set of solutions. Which of the following are always true?

• \(\displaystyle A=B\)
• \(\displaystyle A\) and \(\displaystyle B\) have the same rank
• \(\displaystyle A\) and \(\displaystyle B\) have the same nullity
• \(\displaystyle A=cB\), for some \(\displaystyle c\in \mathbb{R}\)

Hint

Try to construct two simple \(2 \times 2\) matrices that have the same null space.

Answer

• \(\displaystyle A=B\)
• \(\displaystyle A\) and \(\displaystyle B\) have the same rank
• \(\displaystyle A\) and \(\displaystyle B\) have the same nullity
• \(\displaystyle A=cB\), for some \(\displaystyle c\in \mathbb{R}\)

Solution

The set of solutions to \(\displaystyle Ax=0\) is just the null space of \(\displaystyle A\). We see that \(\displaystyle A\) and \(\displaystyle B\) have the same null space. Hence, they have the same nullity. Since both are order \(\displaystyle n\) square matrices, they have the same rank. However, they needn’t be equal or scalar multiples of each other. Here is a simple counter-example:

\[ \begin{equation*} A=\begin{bmatrix} 1 & 1\\ 2 & 2 \end{bmatrix} ,B=\begin{bmatrix} 2 & 2\\ 5 & 5 \end{bmatrix} \end{equation*} \]

The null space for both matrices is \(\displaystyle \text{span}\{( 1,-1)\}\).