Final answer:
The subsets of M₂₂ that are subspaces include upper triangular matrices, skew symmetric matrices, and matrices with a trace of zero. The other given subsets do not satisfy the criteria for a subspace.
Step-by-step explanation:
The question asks which subsets of 2-by-2 matrices (M₂₂) form a subspace of M₂₂. To determine if a subset is a subspace, it must satisfy three criteria: it must contain the zero matrix, be closed under addition, and be closed under scalar multiplication.
Out of the given subsets, the 2-by-2 upper triangular matrices, 2-by-2 skew symmetric matrices, and 2-by-2 matrices with Tr(A)=0 satisfy these criteria and thus are subspaces of M₂₂:
- The set of 2-by-2 upper triangular matrices includes the zero matrix, and the sum of any two upper triangular matrices as well as scalar multiples of an upper triangular matrix are still upper triangular.
- 2-by-2 skew symmetric matrices form a subspace as they too include the zero matrix, and the sum and scalar multiples of skew symmetric matrices are skew symmetric.
- The set of 2-by-2 matrices with Tr(A)=0 is also a subspace since the trace is additive over matrix addition and is unaffected by scalar multiplication.
On the other hand, the sets of 2-by-2 invertible matrices, 2-by-2 singular matrices, 2-by-2 matrices A such that BA = 0₂₂ for a fixed matrix B, 2-by-2 matrices A such that A² = I₂, and 2-by-2 symmetric matrices do not satisfy all of the subspace criteria and are therefore not subspaces of M₂₂.