Suppose A is $n$ x $n$ and the equation A $\vec{x} = \vec{b}$ has a solution for each $\vec{b}$ in $\mathbb{R}^n$

Question

Explain why A must be invertible. Can someone explain why? I am a little confused here.

Answer 1

If it has a solution for each b, (the linear map associated to) $A$ is surjective. But in finite dimensions, for linear maps, the injectivity is equivalent to the surjectivity (it's a consequence of the rank-nullity theorem). Hence the bijectivity, or, in other words, the invertibility of the matrix A

Answer 2

Hint: consider the system $AX=I$. Apply the hypothesis to the columns of $A$.