Consider a matrix $A=(a_{ij})_{ n ×n }$ with integer entries such that $a_{ij}=0$ for $i>j$ and $a_{ii}=1$ for $i=1,…,n$. then which of the followings are true?
$A^{-1}$ exists and it has integer entries.
$A^{-1}$ exists and it has some entries that are not integer.
$A^{-1}$ is a polynomial of $A$ with integer coefficients.
$A^{-1}$ is not a power of $A$ unless $A$ is the identity matrix.
By the given conditions $A$ is the upper triangular matrix with diagonal elements $1$.so eigenvalues are $1$.so their product=determinant of $A =1.$ So 1 is true. Inverse of the identity matrix is itself with has all integer entries so 2 is false. But I have no idea about (3) and (4) can anyone help me please.