TITLE: A matrix equation $Ax=0$ has infinite solutions, does $A^Tx = 0$ have infinite solutions?
QUESTION [4 upvotes]: I'm wondering whether a system with a transpose of a matrix has the same type of solution that the original matrix system has. If an equation $Ax=0$ equation has a unique solution, would a system with $A$ transpose instead of $A$ also have a unique solution? And what about with no solution, and infinite solutions?

REPLY [2 votes]: If you are dealing with square matrices, then the answer is yes. You are essentially asking if the matrix is invertible or not, and $A$ is invertible iff $\det(A)\ne0$ and $\det(A)=\det\!\left(A^T\right)$.
However,
$$
\begin{bmatrix}
3&1&2\\
2&0&1
\end{bmatrix}
\begin{bmatrix}1\\1\\-2
\end{bmatrix}
=\begin{bmatrix}0\\0
\end{bmatrix}
$$
and any real multiple of this solution is a solution. Yet
$$
\begin{bmatrix}
3&2\\
1&0\\
2&1
\end{bmatrix}
x
=\begin{bmatrix}0\\0\\0
\end{bmatrix}
$$
requires $x=\begin{bmatrix}0\\0\end{bmatrix}$.
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