# Suppose $CA=I_n$ (the $n \times n$ identity matrix. Show that the equation $A\textbf{x}=\textbf{0}$ has only the trivial solution [duplicate]

Suppose $$CA=I_n$$ (the $$n \times n$$ identity matrix. Show that the equation $$A\textbf{x}=\textbf{0}$$ has only the trivial solution.

The textbook solution is as follows:

If x satisfies $$A\textbf{x} = \textbf{0}$$, then $$CA\textbf{x} = C\textbf{0} = \textbf{0}$$ and so $$I_n\textbf{x} = \textbf{0}$$ and $$\textbf{x} = \textbf{0}$$. This shows that the equation $$A\textbf{x} = \textbf{0}$$ has no free variables. So every variable is a basic variable and every column of A is a pivot column.

Now I get that $$I_n\textbf{x} = \textbf{0}$$ would mean $$\textbf{x}$$ must be the trivial solution since there are no free variables and you have a pivot position at every column. However, I don't get how that implies that $$A\textbf{x}=\textbf{0}$$ has the trivial solution since $$A \neq I_n$$. I don't quite get how $$C$$ comes into play for solving this problem.

• If $A$ is not invertible, then $A$ cannot be full rank, thus having infinite solutions. I don't know if you know the equivalent formulation of invertiability though. Jan 27, 2021 at 8:30
• So this problem is supposed to be solved without invertibility since it hasn't been covered yet.
– jlcv
Jan 27, 2021 at 8:35
• $I_nx=x$ by definition because $I_n$ is the identity matrix. Thus, $I_nx=0$ and $x=0$ is the same thing. No need to mention free variables or anything else.
– A.Γ.
Jan 27, 2021 at 8:39
• Does this answer your question? Linear Algebra - Suppose $CA=I_n$. Show that the equation $Ax = 0$ has only the trivial solution.
– Ugo
Jul 14, 2021 at 9:28

$$A\mathbf{x} = \mathbf{0}$$

$$\Rightarrow CA\mathbf{x} = C\mathbf{0}$$

$$\Rightarrow I_n\mathbf{x} = C\mathbf{0}$$

$$\Rightarrow \mathbf{x} = C\mathbf{0}$$

$$\Rightarrow \mathbf{x} = \mathbf{0}$$

The second sentence in the texbook solution is incomplete. What they have proved is $$Ax=0$$ implies $$x=0$$ so no vector other than $$0$$ is a solution.

• Sorry I had missed one of the solution sentences, I updated my question to include the missing sentence.
– jlcv
Jan 27, 2021 at 8:34