Question

2. Consider the following set of complex 2 x 2 matrices where i = -1: H = a + bi -c+dil Ic+dia-bi Put B = {1, i, j, k} where = = {[ctdie met di]|1,3,c,dex} 1-[ ), : = [=]. ; = [i -:], « =(: :] . (a) Show that H is a subspace of the real vector space of 2 x 2 matrices with entries from C, that is, show H is closed under matrix addition and multi- plication by a real scalar. (In particular, this shows that His an abelian group with respect to addition.) (b) Show that B is a basis for H, regarded as a real vector space. (Thus H is a four-dimensional real vector space.) (c) Let A EH. Prove that A is invertible if A is not the zero matrix, and find a formula for A-. Deduce that H is closed under taking inverses of nonzero elements. (d) Observe also that H has the following equivalent description: H = {[%: 7] |zwec}. 22 Let A, B E H so that A = 21 -22 and B _wi-W2 = -2, for some W2 W1 21, 22, w1, 2 E C. Find a formula for the matrix product AB, in terms of 21, 22, w1, 2, and deduce that H is closed under matrix multiplication. (You may use standard facts about complex conjugation.) (In particular, together with part (a), it follows from this that H is an arith- metic with addition and multiplication that satisfies all of the conditions that define a field, except for commutativity of multiplication. Thus H forms an algebraic system known as a skew field or a division ring.) (e) Verify the following equations: i = ja = k = ijk = -1. (It follows, or you can check directly, that ij=-ji=k,jk = -kj=i and ki -ik-j, so certainly the multiplication in H is not commutative.) (You have just followed in the footsteps of William Rowan Hamilton, who carved these equations, since become famous, in stone on Broom Bridge in Dublin in 1843. This was the culmination of Hamilton's quest to find analogues of R and C in dimensions higher than two. In fact, no division ring exists as a real vector space of dimension three.)

Answer 1

2 HS laebi -et dill Predi a-bila, bolol, ERG @ we have to prove that His closed under matrix adalition and multiplication by real scalar A = (atbi ut diil - lata i ap-bli i B= (a + bal -(dt dail I Cat dai cd-bai , aziba, ca, do fr A+B= (atadt (bitha) i -(4+2) + Girder 1 etla + (autoe) i Cartad) - hitha) i there autod betha, atlas de toda all are Real number (: 'a', b, c, di fR, itd) Hence His closed under matrix addition LA = ( Lastabii nag tadri la ata dei day-abli day, 261, 24, adi are real runter. ( Real number is closed unden scalar multiplication) Hence His closed under scalar multi- plication where the seakan is a real nunher. Hence (a) is proved

@ .= $1, ;, iss. 1+ (60).:-()=688), 4() B Here 1,; , j and is are che anley independent. mlow H= Catib nee di labeled fr letai arbre - dim (H) = 4 CA tid 5966954c (31)-%) 463 ;) = 9(1) tec7)+b6j) + d(re) Hence all element of His unean combination of elements of B. and Bis & crearly independent element thene B is a basis of H.

Now Att Alathi act dil codi arbi det A = (atbi) (a-bi), Fethi) (erdi) - (athi) Carbi) + (ca di) (et li) - atb? +224 de So det it can not he zero entil A when aFbFe=d-o then def Aso. then a cool so AE t is inuentible for all values of albie, der encept a h-e-d-o so A B not invertible when A= Cool alow A=(2, -za , whene 21 = atbi za 2 22= etdi det A: 21 : 2) + 222 =213 +122). A p till ut Als & stit.

Here p , Q are both on-singulan - entept asb =(=doto. so both p and Q are innertible Antal = 87 A. (p-tia) (stit) = 1 tio. PS- QJ ti (PT + Qs) - Irio. ps-QT -l. A PT + Qs = 0. PT = -QS. DS- Q . ora 11= - plast 7.PS- &(-plas) = 1. pst aptus si -> (P + apl@)s = ? > s= (p te pt al. Adept

Tarpt & (pt apta) A7 = (8 + 9p74) 7j pla (pt aplan where p. ( a c) , a=(2, 4) Then you can eaisly find at.