(2) (a) For any O E [ 0 21] let -sino Cose x For Cosce sino 1² [ a b ] simplity any matrix A АХ 052 If A = and [33]... B =[2] C], find X-sored that A(x+B) = C. Q 2 (C) Let S be the set of matrices of the form As a a2 ag where arbitrary real numbers. Show there exists a unique matrix E in s such that A EA for all o in وگرنه...
Q 2 (c) Let S be the set of matrices of the form A = a, a T ag arbitrosy where are real numbers. Show there exists a unique matrix E in s such that АЕА o in S. for all Marks ((1+3+37 +(2+3 + 8) = 20 Marks) MATH 2118 Online Class Exercise I Qla) Sketch the surface s defined by the equation z = =9-6tty! (6) Determine the equation of the tongent plane to the surface s given...
of the the be Let s set of matrice's sa, az - form A= where are a,, az Lo a 3 real numbers. Show there exish unique matrix E ins such that AE = A for all А S arbitary real in
Show that the set of matrices of the form
where a, b ∈ Q is a field under the operations of matrix addition
and multiplication. (abstract algebra)
please show the following axioms (closure, identity,
associative, distributive, inverse, and commutative) for addition
and multiplication
a 6 26 a
LO 2a 4) Let V be the set of diagonal 2x2 matrices of the form la ). Determine whether or not this set is a subspace of the set of all real-valued 2x2 matrices, M22, with standard matrix addition and scalar multiplication. Justify your answer.
Let M be the set of 2 x 2 matrices of the form (82) where a, d ER-{0}. Consider the usual matrix multiplication, i.e: ae + bg af +bh ce + dg cf + dh (2)) = (ce ) (a) Show that (M,-) is an abelian group. (b) Compute the cyclic subgroup generated by M = What is the order of M? (6 -4) € M.
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are similar over C, then they are also similar over IK (a) Prove that for all X, y є Rnxn, the function f(t) det (X + ty) is a polynomial in t. (b) Prove that if X and Y are real n × n matrices such that X + ừ is an invertible complex matrix, then there exists a...
4. Let M be the set of 2 x 2 matrices of the form (62) where a, d E R - {0}. Consider the usual matrix multiplication ·, i.e: ae + bg af + bh ce + dg cf + dh (a) Show that (M,·) is an abelian group. 1 (b) Compute the cyclic subgroup generated by M = What is the order of M? 66 -4) (1) EM EM.
#21. Let G be the set of all real 2 x 2 matrices where ad + 0, Prove that under matrix multiplication. Let N = (a) N is a normal subgroup of G. (b) G/N is abelian.
Let S be the set/vector space of all real numbers of the form a sart(2)+ b'pi, where a, b are any real numbers, where we add these numbers the usual way, and multiply by real number scalars the usual way. Find, another, simpler way, of describing this vector space