Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places

Question

Let $a>0$. Let $A$ be the $n\times n$ matrix with $a+1$ on the diagonal and $a$ in all other entries. How can one compute $A^{-1}$ as a function of $n$?

Answer 1

Write $A$ as $I + aJ$ where $J$ is the matrix consisting of all $1$s. Now using the fact that $J^2 = nJ$, look for an inverse of the form $c_1I + c_2J$. More generally, there is the famous Sherman-Morrison formula for rank-one perturbations of invertible matrices.

Answer 2

Let $J$ be an $n$ by$n$ matrix all whose entries are 1. Then the answer is $I-aJ/(1+an)$. as you can verify. To motivate and get to this formula look up a series expansion for $(I-A)^{-1}$.

Answer 3

Write

$$A = I_n + a 1_n 1_n^T$$

Using Sherman-Morrison,

$$A^{-1} = I_n - \left(\frac{a}{1 + a 1_n^T 1_n}\right) 1_n 1_n^T = I_n - \left(\frac{a}{1 + n a}\right) 1_n 1_n^T$$