Which of the following sets are closed under scalar multipli

Which of the following sets are closed under scalar multiplication? (i)The set of all vectors in R^2 of the form (a, b) where a - 5b = 0. (ii)The set of all 3 times 3 matrices whose trace is equal to 4. (iii)The set of all polynomials in P_2 of the form a_0 + a_1 x + a_2 x^2 where the product a_0 a_1 a_2 >orequalto 0. none of them (ii) and (iii) only (i) and (iii) only (ii) only (iii) only (i) only all of them (i) and (ii) only

Solution

1) a - 5b = 0

2 ( a,b) = 2( a- 5b) = 2* 0

2a - 10b = 0

hence closed under scalar multiplication

set of 3x3 matrix whose trace is equal to 4

let trace ( A) = m11 + m22 + m33 = 4

trace ( Am ) = A m11 + A m22 + A m33 is not equal to 4

hence , nlot closed underscalar multipolication

3) set of all polynomials in P2 of the form a0 +a1x + a2x^2

where product a1 a2 a3 >=0

atleast one coefficient a1 , a2 , a3>=0

this is closed under scalar multiplication

hence option C is correct