Existence of Solution of a matrix equation

Question

Suppose, $Ax=b$ does not admit any solution for some $n*n$ matrix & some b in $R^n$.Does it imply $A^tx=b$ also not admit solution ?

I think it may admit solution but I am unable to find out counter example. pls anyone help me. Thanks in Advance

Answer 1

Counterexample:

Consider the linear system $$\begin{bmatrix} -1 & -2\\ -1 & -2 \end{bmatrix} \begin{bmatrix} x\\ y\end{bmatrix}=\begin{bmatrix} 1 \\ 2\end{bmatrix}$$ for which there isn't solution, that is, is inconsistent.

However, the linear system $$\begin{bmatrix} -1 & -2\\ -1 & -2 \end{bmatrix}^{T} \begin{bmatrix} x\\ y\end{bmatrix}=\begin{bmatrix} 1 \\ 2\end{bmatrix} \text{equivalent to} \quad \begin{bmatrix} -1 & -1\\ -2 & -2 \end{bmatrix}^{T} \begin{bmatrix} x\\ y\end{bmatrix}=\begin{bmatrix} 1 \\ 2\end{bmatrix}$$ has infinitely many solutions, that is, is consistent.