when A matrix has one solution IF M>N

Question

if a matrix A is m by n , and m>n , in this case we have one solution? i don't understand how more rows of a matrix can still have one solution

Answer 1

Consider $$ A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} $$ Then $Ax = 0$ has exactly one solution (namely $\begin{bmatrix}0\\0 \end{bmatrix}$). Is that what you were looking for?

Answer 2

I’m assuming you are referring to the equation $Ax = b$. In the case that $A \in \mathbb R^{m \times n}, x \in \mathbb R^n,$ and $b \in \mathbb R^m$, where $m > n$, then whether there exists an $x$ that satisfies this equation will depend on the elements of $A$ and $b$. For example, suppose that the bottom $m - n$ rows of both $A$ and $b$ are all $0$. You can then solve for $x$ by considering only the top $n$ rows of $A$ and $b$.