Let X= [ array cc 1 & -1 1 & 1 array ] , Let Y be a 2 2 real matrix satisfying the condition X Y=Y X . Then the smallest possible value of det (Y) is

Step-by-step solution

X= [ array cc 1 & -1 1 & 1 array ], Y_ 2 2 =?, X Y=Y X Let Y= [ array ll x & y z & t array ] such that X Y=Y Z [ array cc 1 & -1 1 & 1 array ] [ array ll x & y z & t array ]= [ array cc x & y z & t array ] [ array cc 1 & -1 1 & 1 array ] [ array ll x-z & y-t x+z & y+t array ]= [ array ll x+y & -x+y z+t & -z+t array ] x+z=z+t,y-t=-x+y, x-z=x+y x=t y=-zY= [ array cc t & -z z & t array ] |Y|=t^2+z^2 which is always non-negative for t, Z R . Smallest value of |Y|=0