Let P be a 2 2 matrix such that P [ array c 1 -1 array ]= [ array c -1 2 array ] and P^2 [ array c 1 -1 array ]= [ array l 1 0 array ] . If x₁ and x₂ are two values of x for which |P-x I|=0 , where I is an identity matrix of order 2 then find the value of x₁^2+x₂^2 .

Step-by-step solution

Let aligned P & = [ array ll a & b c & d array ] P [ array c 1 -1 array ] & = [ array ll a & b c & d array ] [ array c 1 -1 array ]= [ array l a-b c-d array ]= [ array c -1 2 array ] aligned -a-b=-1 ...(1) c-d=2 ...(2) and P P [ array c 1 -1 array ]=P [ array c -1 2 array ]= [ array l 1 0 array ] -a+2 b=1 ...(3) and -c+2 d=0 ...(4) From (1) and (3), b=0 and a=-1 From (2) and (4), d=2 and c=4 aligned & P= [ array cc -1 & 0 4 & 2 array ] & |P=x I|= [ array cc -1-x & 0 4 & 2-x array ]=0 & (x+1)(x-2)=0 x=-1,2 & x₁^2+x₂