Let $A$ be such a 4×4 matrix that the equation $Ax = (1, 0, 0, 0)$ has a unique solution $x$. Is $A$ necessarily invertible?
I know that if A is invertible, the equation $Ax = y$ has for all $y \in \mathbb R^n$ a unique solution $x = A^{-1} y \in \mathbb R^n$. But there's only one unique solution, could someone talk me through this?
Best Answer
Have you studied the algorithm to compute the general solution of a linear system? It should come quite early in a linear algebra course (for instance, in my course it came before the definition of a matrix inverse). It is an easy consequence of that algorithm that if a square matrix $A$ is not invertible then for each $b$ the equation $Ax=b$ has either no solutions or an infinite number of them (in $\mathbb{R}^n$).