Decide whether the given statement is true or false. If the statement is false, give a counterexample;?

if the statement is true, give a proof.





(a) If the eigenvalues of a 2 脳 2 matrix are equal to 1, then the four entries of that matrix are each less than 500.





(b) The trace of the product of two n 脳 n matrices is the product of the traces.Decide whether the given statement is true or false. If the statement is false, give a counterexample;?
(b) The trace of the product of two n 脳 n matrices is the product of the traces


False


A = 1 2


------3 4


B = 3 4


------5 6


AB = 13 16


--------29 36


Trace (A) = 5


Trace(B) = 9


Trace (AB) = 49


(5)(9) is 鈮?49Decide whether the given statement is true or false. If the statement is false, give a counterexample;?
(a) is false


for the eigenvalues to be the same 4bc+(a-d)^2 must equal zero


for the eigenvalues to both be 1 we also need a+d = 2


so for example a=d=1 will work whenever bc = 0


b=0 c=501





(b) is false


(a+b)(e+h) 鈮?ae+bg+cf+dh