Question

Let `p` be a non singular matrix, and `I + P + p^2 + ... + p^n = 0,` then find `p^-1`.

Answer 1

We have, `I+p++p^(2)+...+p^(n)=O` ...(1)
Since p is non-sigular matrix, p is invertible. Multiplying both sides of (1) by `p^(-1)`, we get
`implies =^(-1)I+p^(-1)p+p^(-1)p^(2)+...+p^(-1) p^(n)=p^(-1)O`
`implies p^(-1)+I+p+p^(2)+...+p^(n-1)=O`
`implies p^(-1)=-(I+p+p^(2)+...+p^(n-1))`
`implies p^(-1)=- (-p^(n))=p^(n)`