Question

Which of the following subsets in M₂₂ is a subspace of M₂₂: A. The set of all matrices A with det (A) = 0. B. The set of all matrices A with det (A) = 1. C. The set of all matrices A with tr (A) = 0, where tr (A) represents the trace of matrix A. D. The set of all matrices A with all diagonal elements equal to 1 and all other elements equal to 0.

Answer 1

Only Option C, the set of all matrices A with tr (A) = 0, is a subspace of M₂₂ because it is closed under addition, scalar multiplication, and contains the zero matrix.

Step-by-step explanation:

The question refers to determining which subset of matrices M₂₂ (the set of all 2x2 matrices) forms a subspace. A subspace must be closed under addition and scalar multiplication and contain the zero vector (the zero matrix in this case).

Option A specifies the set of all matrices with determinant zero. This set is not closed under scalar multiplication because scaling a non-zero matrix with determinant zero can result in a matrix with a non-zero determinant.

Option B, the set of matrices with determinant equal to one, is not a subspace because it is not closed under addition or scalar multiplication; the sum or scalar multiple of two such matrices does not generally have determinant one.

Option C, which includes matrices with a trace of zero, is a subspace of M₂₂. The trace is a linear function, meaning the sum of two matrices with zero trace also has a zero trace, and scalar multiplication of a matrix with zero trace by any scalar also results in a matrix with zero trace. Additionally, the zero matrix is included in this set.

Option D specifies matrices that are a particular form and not closed under addition or scalar multiplication. As such, this set is also not a subspace.