Question

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

Answer 1

Let:

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

If `\ Tr(bb(A)) = 0`. then:

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

Then for any real number `mu`, we have:

`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 `2xx2` matrix with trace `0` is closed under scaler multiplication QED.