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Prove that if m is an odd number then ,4 is singular. Prove also that if A has an inverse then the inverse is a skew symmetric matrix. 11. Let 3? be the field of real numbers and ZFm the total m by m matrix ring over 8F. Prove that for every element X and Y of ^m if XY — YX commutes with X then XY - YX is nilpotent. 12. Let X and Y be m by m matrices over the field of real numbers and Im the m by m unity matrix. Prove that XY— YX — Im 4= 0. 13. Let N be a nilpotent element of a total m by m matrix ring ^m where OF is a field.

Let 3? be the field of real numbers and ZFm the total m by m matrix ring over 8F. Prove that for every element X and Y of ^m if XY — YX commutes with X then XY - YX is nilpotent. 12. Let X and Y be m by m matrices over the field of real numbers and Im the m by m unity matrix. Prove that XY— YX — Im 4= 0. 13. Let N be a nilpotent element of a total m by m matrix ring ^m where OF is a field. Prove that N + / m (where / m is the m by m unity matrix) is a nonsingular element of J^ m . 14. Let ^ be the set of all 2 by 2 matrices (atj) over the field of real numbers such that = a = — #11 #22 *"ld #12 #21- Prove that %> with the usual matrix addition and multiplication is a field and is isomorphic to the field of all complex numbers.

T=i THEOREM 4. L^r TX,V2,... ,Vn be subspaces then (18) through (20) are equivalent statements. of a vector space T (18) r = ^rt. T=jtrt and %Vi = 0 with VieTi (19) implies V{ = Ofor i = 1, 2 , . . , n. r = iri i=l and r i n s r j = {0} for /= 1,2 «. (20) i =*= i Proo/. Let (18) hold. Then clearly, T = 2 ^ i and 2 ^ = 0 implies K* = 0 since, otherwise, 0 will have two distinct representations. Thus, (19) is established. Next, let (19) hold and let a nonzero element Ut be such that U{ e T{ and Ui = 2 Vj with V5 e T5.

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