Let $A=(a_{ij})_{n\times n}$ be a invertible matrix with the following property $$ a_{ij}=0,\quad \text{when}\,\,\,i+j\,\,\text{is odd}. $$ My question: Does the inverse matrix $A^{-1}=(b_{ij})_{n\times n}$ also have this property, i.e., $b_{ij}=0,\quad \text{when}\,\,\,i+j\,\,\text{is odd}$?
This can be viewd as a analogue of the fact that the inverse of a invertible upper triangular matrix is also upper triangular. It's also thue for $n=2,3$ by direct computation, and I think it holds for general $n$ by using the cofactor, but I'm not sure if if works and if there are better way to prove this.
As user1551's comment states, an $n \times n$ matrix $A$ with this property can be made block-diagonal with a suitable permutation similarity. In particular, there exists a permutation matrix $P$ such that $A$ has this pattern of entries if and only if $P^TAP$ is block diagonal with block sizes $n/2$ and $n/2$ where $n$ is even and $(n+1)/2, (n-1)/2$ where $n$ is odd. With this established, it suffices to note that this block-diagonal structure is preserved when the inverse of a matrix is taken.
In particular, if we have $$ P^TAP = \pmatrix{M_1 & 0\\0 & M_2} \iff A = P\pmatrix{M_1 & 0\\0 & M_2}P^T, $$ with $M_1,M_2$ square, then it follows that $$ (P^TAP)^{-1} = P^TA^{-1}P = \pmatrix{M_1^{-1} & 0\\0 & M_2^{-1}}, $$ so that $$ A^{-1} = P\pmatrix{M_1^{-1} & 0\\0 & M_2^{-1}}P^T, $$ ensuring that $A^{-1}$ has the expected pattern of zero-entries.
As a bit of a hint towards the details, here's what this looks like in the $5 \times 5$ case. Consider $$ A = \pmatrix{ a_{11} & 0 & a_{13} & 0 & a_{15}\\ 0&a_{22} & 0 & a_{24} & 0\\ a_{31} & 0 & a_{33} & 0 & a_{35}\\ 0&a_{42} & 0 & a_{44} & 0\\ a_{51} & 0 & a_{53} & 0 & a_{55}}, \quad P = \pmatrix{ 1&0&0&0&0\\ 0&0&0&1&0\\ 0&1&0&0&0\\ 0&0&0&0&1\\ 0&0&1&0&0}. $$ Pay particular attention to the order of the columns of $P$. You should find that $$ P^TAP = \pmatrix{ a_{11} & a_{13} & a_{15} & 0 & 0\\ a_{31} & a_{33} & a_{35} & 0 & 0\\ a_{51} & a_{53} & a_{55} & 0 & 0\\ 0&0&0&a_{22} & a_{24}\\ 0&0&0&a_{42} & a_{44} }. $$