Question

Question 5) (8 points) Consider the following subset S = {A € M3(R): AT = A, and every diagonal element of A is 0} (In words, S is the set of all symmetric 3 x 3 matrices that all have all O's on their diagonal) (a) Prove that S is a subspace of M3(R) (b) Determine a basis for S and state the dimension of S

Answer 1

Solution (5) ► Griven that s={ AE Mz(R) : ATEA ,ond every diagonal element of A is o} proof ca). is clearly to :]es element Beszero 8 Atte"- A+B, every diagonal element o i e zero vector belong in s (11) Let A ,BES such that A,B EM3C0R) ef A1BES → ATAond every diagonal of A es zero +B+=B, and every diagonal element of Now A+B, and of A+B) is o (A+B) I A+B, and evry diagonal element of (A+B) es [A+BES dii) let del and AES diagonal if AES » AT=A, and every element of Aes o QAT-QA , and every diagonal element of LA iso element (OCA) I XA, and every > CAES from condition ;) (W) and (in) clearly can say that ses a subspace of M₂ (R) 80 diagonal of (QA) es o we

is (6) Griven matix How Peymonetic je 013 qui a12 9,2 922 ie AT= 923 A a 3 423 0133 3x3 Here Griven that all =922 = 933 = 0 we get a 12 913 012 O 923 A- a23 O 913 beiven Mahix can be written as 0 0 1 OI o 1 o + 013 109/2010 0 0 0 A= a 12 1 I o 0 0 0 9 Henee Basis of s=d 7481031) (3:] element in and dim(s) total number of Basis of s din (s)=3 AR