(e) Let B= ( a b c d be any 2 x 2 matrix. а cos a] (i) Show that there are real numbers U11 and a such that al = U11 sin

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(e) Let B= ( a b c d be any 2 x 2 matrix. а cos a] (i) Show that there are real numbers U11 and a such that al = U11 sin α (ii) Let a E R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that COS a - sin al [国 = U12 + U22 d sin a COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RAU = for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RQU Ꭱ = RØV, where a, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = +V. - = ។