Must it be true that A = Ra, for some a € R? Either prove this, or give a counterexample (including justification). (e)

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Must It Be True That A Ra For Some A R Either Prove This Or Give A Counterexample Including Justification E 1 (95.61 KiB) Viewed 34 times

Must it be true that A = Ra, for some a € R? Either prove this, or give a counterexample (including justification). (e) Let B [a b] be any 2 x 2 matrix. (i) Show that there are real numbers u₁₁ and a such that [a] - = U11 COS a sin a (ii) Let a € R. Use the invertibility of Ra to prove that there are unique U12, U22 ER such that COS a sin a [] = U12 +422 sin a COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B RQU for some a ER and some upper-triangular matrix U. (iv) Suppose that B R&U= RV, where a, ß E R and U and V are upper- triangular. Prove that if B is invertible, then U = IV. -