Let A= [ array cc 0 & 2 -2 & 0 array ] and (m I+n A)^2=A where m, n are positive real numbers and I is the identity matrix. What is (m+n) equal to?

Step-by-step solution

aligned & A= [ array cc 0 & 2 -2 & 0 array ] & MI m [ array ll 1 & 0 0 & 1 array ]= [ array cc m & 0 0 & m array ] & n ~A n [ array cc 0 & 2 -2 & 0 array ]= [ array cc 0 & 2 n -2 n & 0 array ] & Here, MI +n ~A [ array cc m & 0 0 & m array ]+ [ array cc 0 & 2 n -2 n & 0 array ]= [ array cc m & 2 n -2 n & m array ] & Given ( MI +n ~A )^2 & ~A = [ array cc m & 2 n -2 n & m array ] [ array cc m & 2 n -2 n & m array ]= [ array cc 0 & 2 -2 & 0 array ] & 4 m n=2 and m^2-4 n^2=0 & m+n=1+ 1 2 = 3 2 aligned