Question

8. Let Maxn denote the vector space of all n x n matrices. a. Let S C Max denote the set of symmetric matrices (those satisfying AT = A). Show that S is a subspace of Mx. What is its dimension? b. Let KC Maxn denote the set of skew-symmetric matrices (those satisfying A' = -A). Show that K is a subspace of Max. What is its dimension?

Answer 1

8. (a) Since 8c Mnxn is the set of symmetric matrices so 8=2 A EMnenia a A} het A,BE Ô Then As A and BT=B nos (A+B) - Atom = A+BIASA BEB7 → (A+B)ES Let CER Now (CA)* = CART = CA TAT=A] CAE 8 This A.BE & > A+BES and ce IR and AE ô » CAE 8. This proves that s is a subspace of Mnxn. The dimension of ois (n2n +n) 7 + +1) 6 since I K c Mnan is the set of sken symmetric matrices so x=2A EMnen: At= - A} Let A, BE X X then A=-A and Bob Nan A+B)T= AT+BT -A-B -- A+B) > A+BEK

Let CFR non (CA)TE CAT 1 = c(-A) s-CA * CAFK This A. BE K * At Bek and cell and Atk » CAFK This prooves that k is a subspace of Mnen and dimension of k = n(n-a)