Let P be a 3x3 non null matrix with real numbers such that P^3=O where O is the null matrix of 3x3, then prove the following.(A,B,C)

Question

A) det of (4P^2 -2P +I) is non zero number, where I is the 3x3 identity matrix

B) I-2P is an invertible matrix,, where I is the 3x3 identity matrix

C)If matrix P has integer entries then the absolute value of the det of (I-4P^2)=1 , , where I is the 3x3 identity matrix.

I don't even know how to proceed in this question, the only thing i can think of is that if I factor some AB=I matrix then take det on both sides , detA detB =1 then i can prove thatthey are non zero for the product to be =1

Answer 1

A) Since $P^3$=0, P is nilpotent. Hence its eigenvalues are 0,0,0. Hence eigenvalues of A=$4P^2$ -2P +I are 1,1,1. Hence determinant of A= Product of its eigenvalues =1. B) Eigenvalues of I-2P are also 1,1,1. Hence it’s invertible. C) I-$4P^2$ has eigenvalues 1,1,1. Hence its det is 1.