Question

Suppose $CA=I_n$(the $n\times n$ identity matrix). Show that the equation $Ax=0$ has only the trivial solution. Explain why Acannot have more columns than rows.

Short answer

Amust have at least as many rows as columns because each pivot is in a different row.

Step-by-step solution

The equation $Ax=0$ has the trivial solution

When $x$ satisfies the equation $Ax=0$, then $CA\mathrm{x}=C\times 0=0$. So, $I_n\mathrm{x}=0$ and $\mathrm{x}=0$. This demonstrates that there are no free variables in the equation $Ax=0$.

Explanation for A cannot have more columns than rows

Therefore, every variable is a basic variable since the equation $Ax=0$ has no free variables. Each column of Ais a pivot column. Thus, Amust have at least as many rows as columns because each pivot is in a different row.