Suppose\(A\) is the \(3 \times 3\) zero matrix (with all zero entries). Describe the solution set of the equation \(Ax = 0\).

Write matrix \(A\) as an augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right].\)

\(\left[ {\begin{array}{*{20}{c}}0&0&0&0\\0&0&0&0\\0&0&0&0\end{array}} \right]\)

The variables corresponding to pivot columns in the matrix are called basic variables.The other variable is called a free variable.

Here, \({x_1},{x_2}\), and \({x_3}\) are free variables.

The general solution of the system is:

\(\begin{aligned}{c}x = \left[ {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right]\\ = {x_1}\left[ {\begin{aligned}{*{20}{c}}1\\0\\0\end{aligned}} \right] + {x_2}\left[ {\begin{aligned}{*{20}{c}}0\\1\\0\end{aligned}} \right] + {x_3}\left[ {\begin{aligned}{*{20}{c}}0\\0\\1\end{aligned}} \right]\end{aligned}\)

Each \(x\) in \({\mathbb{R}^3}\) satisfies \(Ax = 0\) if \(A\) is a \(3 \times 3\) zero matrix. Therefore, the solution set of the equation \(Ax = 0\) consists of all vectors in \({\mathbb{R}^3}\).