Erickson Bartlett

05/27/2023 · Junior High School

2. Use Theorem 4.2 .1 to determine which of the following are subspaces of \( M_{n n} \). (a) The set of all diagonal \( n \times n \) matrices. (b) The set of all \( n \times n \) matrices \( A \) such that \( \operatorname{det}(A)=0 \). (c) The set of all \( n \times n \) matrices \( A \) such that \( \operatorname{tr}(A)=0 \). (d) The set of all symmetric \( n \times n \) matrices. (e) The set of all \( n \times n \) matrices \( A \) such that \( A^{T}=-A \). (f) The set of all \( n \times n \) matrices \( A \) for which \( A \mathbf{x}=0 \) has only the trivial solution. (g) The set of all \( n \times n \) matrices \( A \) such that \( A B=B A \) for some fixed \( n \times n \) matrix \( B \).