(1 point) The trace of a square n × n matrix A = (aj) is the sum A1 + a22 + + an
ID: 3136214 • Letter: #
Question
(1 point) The trace of a square n × n matrix A = (aj) is the sum A1 + a22 + + ann of the entries on its main diagonal. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 matrices with real entries that have trace 1. Is H a subspace of the vector space V? 1. Is H nonempty? H is nonempty 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a 1 21 [5 6 comma separated list and syntax such as [[1,2],[3,4]], [[5,6],[7,8]] for the answer (Hint: to show that H is not closed under addition, it is sufficient to find two trace one matrices A and B such that A B has trace not equal to one.)Explanation / Answer
a
b
c
1-a
where a,b c are arbitrary real numbers. Apparently, H is non-empty.
p
q
r
1-p
be another arbitrary element of H, where p,q,r are arbitrary real numbers. Then A+B =
a+p
b+q
c+r
2-a-p
The trace of A+B is a+p+2-a-p = 2. Hence A+B H so that H is not closed under addition.
a
b
c
1-a
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