Let n be a positive integer and let denotes the space of all n x n real matrices. If is a linear transformation such thatT(A) = 0 whenever is symmetric or skew symmetric then the rank of T is

a)

b)
n²

c)
n

d)
0

Correct answer is option 'D'. Can you explain this answer?

Question

Let n be a positive integer and let

denotes the space of all n x n real matrices. If

is a linear transformation such thatT(A) = 0 whenever

is symmetric or skew symmetric then the rank of T is

a)
b)
n²
c)
n
d)
0

Correct answer is option 'D'. Can you explain this answer?

Answer

I think as we have transformation such that T(A)=0 where A is symmetric or skewnow as every metrix can be written in the sum of symmetric and skew symmetric so T(A)=T((A+A')/2 +( A-A' )/2 ) = T((A+A'/2)) +T((A-A')/2) as T is linear transformationso T(A) =0 this implies this transformation sending every metrix to 0 so it's null space contains all the metrix so Null space become Mn(R) whose dimension is nso by rank nullity theorem, nullity+ rank = dimensionsso dimension of range space is 0 so it's rank is 0

Answer

D

Answer

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