Consider a matrix$ A = (a_{ij})_{n\times n}$ with integer entries such that $a_{ij} = 0$ for $i>j$ and $a_{ii} = 1$ for $i=1\dots n$. Which of the following properties must be true?
- $A^{-1}$ exists and it has integer entries
- $A^{-1}$ exists and it has some entries which are nt integers
- $A^{-1}$ is a polynomial function of $A$ with integer coefficients
- $A^{-1}$ is not a power of $A$ unless $A$ is an identity matrix
I am confuse about fourth option.