Is there an explicit formula for $\left(xx^T\right)^{-1}$ with $x\in\mathbb{R}^n\setminus\left\{0\right\}$?

Question

Let $x\in\mathbb{R}^n\setminus\left\{0\right\}$. Obviously, $$A:=xx^T$$ is symmetric and positive definite. Hence, $A$ is invertible. Can we find an explicit formula for $A^{-1}$?

Answer 1

Obviously the matrix $xx^T$ is not positive definite if $n>1$, because it has rank $1$.

A simple way to see it has rank $1$ is by observing that any vector $y$ which is orthogonal to $x$ belongs to the null space of $A=xx^T$.