Question

3. Let Un (R) be the subgroup of GLn(R) consisting of upper triangular matrices and let Dn(R) be the subgroup of GLn(R) consisting of diagonal matrices. (a) Show that \ : Un(R) + Dn(R), A + diag(a11, ..., ann) is a homomorphism of groups. Find the kernel and image of y. (b) Let Zn(R) = {aIn : a E R*} be the subgroup of Dn(R) consisting of scalar matrices. Determine x-1(Zn(R)). Justify your answers.

Answer 1

(a) 4: UncIR) - Dn(IR) A diag (an, arr. –, ann) Ul win: 4 is group homomorphism. het A, B be two upper triangular non matrices ABE UNOR) ; when A= B= is ; (ai;) (bij) aij = bij=o when when So, A+B= (aijtbii) = (c) Cii -o ر< 4 (A+B) - WIKI;)) ding (c.1, 22, ..., can) = = diag (9,1+bin, azzt bzz ...-., ann+ban) - diag (an, arr. ... ann) + diag (b,,, b22,-., ban = V(A) + 4CB). Also, and So, zero non identity & is natrix is mapped to In matrix is mapped to group homomorphism. wo diagonal matix In.

kery - At Un CIR) 4 (A) = In 3 = { A E Un (12) , diag (an, 922,-., and) - In} - {AE Un (IR) , akk=1 & KE{1,2,...,n3} - Al A= (ais) where a;;= o if iss i=j 99 = 1 if Image 4 = { 4 (A) I AE UN (18)} = { dias (ain, 922, .., annd} = all non diagonal matrices with real enties U (IR)= a In 1 ac 12 * 3. 4t ( 241 (12)) = & A EUN (IR) | 4 (A) E UN (IR)} { A E Un (12) I diap(a.,.. , ann) = ain} - where O & AE UN (IR) | A = (ais) where aig = o is; ajj = a i=j ає Ір

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