Trick for Inverse Hollow Matrix Calculation (Self-Answered)

Question

Let $A$ be the hollow matrix : $$ A=\begin{pmatrix} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 \end{pmatrix} $$

Find the inverse matrix $A^{-1}$ without using any elementary row/ column operations.

Answer 1

Notice that $$ A= \begin{pmatrix} 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1 \end{pmatrix} - I_{4\times4} $$

Let $$ B= \begin{pmatrix} 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1 \end{pmatrix} $$

then $B=A+I$. Observe that $B^2 = (A+I)^2=4B$. Expanding, $$ A^2+2A+I = 4(A+I) \implies A^2-2A=3I $$ Therefore $A^{-1}=\frac{1}{3}(A-2I)$.

More generally, this method gives us an easy way to express the inverse of a hollow matrix $A$ (of the particular stated form) of any order $n$, which is the main reason for posting this question.