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In Exercises 3-4, use the Subspace Test to determine which of the sets are subspaces of Mnn.


In Exercises 3-4, use the Subspace Test to determine which of the sets are subspaces of Mnn.

 3. a. The set of all diagonal n x n matrices.

 b. The set of all n × n matrices A such that det(A) = 0.

 c. The set of all n × n matrices A such that tr(A) = 0.

 d. The set of all symmetric n × n matrices.


4.

 a. The set of all n × n matrices A such that AT = -A.

 b. The set of all n × n matrices A for which Ax = 0 has only the trivial solution.

 c. The set of all n × n matrices A such that AB = BA for some fixed n × n matrix B.

 d. The set of all invertible n × n matrices.

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