Question

Is the set of all 2x2 matrices with zero trace (the sum of diagonal elements) a subspace of all 2x2 matrices (denoted as M₂)?

Answer 1

Yes, the set of all 2x2 matrices with zero trace is a subspace of all 2x2 matrices.

Step-by-step explanation:

Yes, the set of all 2x2 matrices with zero trace is a subspace of all 2x2 matrices.

A subspace is a subset of a vector space that is closed under addition and scalar multiplication. In this case, let's denote the set of all 2x2 matrices with zero trace as S.

1. S is closed under addition because if A and B are matrices in S, then the sum of their traces is zero. So, (A + B) also has a trace of zero.
2. S is closed under scalar multiplication because if A is a matrix in S and c is a scalar, then the trace of cA is c times the trace of A, which is still zero.

Therefore, S satisfies the conditions to be a subspace, and it is a subspace of all 2x2 matrices (M₂).