# Is the set of all #2 × 2# matrices whose trace is equal to #0# closed under scalar multiplication?

##### 1 Answer

Sep 24, 2017

Let:

# bb(A) = ( (a_11, a_12), (a_21, a_22) ) #

If

# a_11+a_22 = 0 # ..... [A]

Then for any real number

# mu bb(A) = mu ( (a_11, a_12), (a_21, a_22) )#

# \ \ \ \ \ = ( (mua_11, mua_12), (mua_21, mua_22) )#

And so:

# Tr(mu bb(A)) = mua_11 + mua_22 #

# \ \ \ \ \ \ \ \ \ \ \ \ \ = mu(a_11 + a_22) #

# \ \ \ \ \ \ \ \ \ \ \ \ \ = mu xx 0 \ \ \ # by [A]

# \ \ \ \ \ \ \ \ \ \ \ \ \ = 0 #

Hence, a