(e) Let B =be any 2 x 2 matrix.COS a(i) Show that there are real numbers un and a such thatu11sin aHint: erpressas a scalar multiple of a unit vector, and hence find anerpression for un in terms of a and c.(ii) Let a € R. Use the invertibility of R, to prove that there are uniqueu12, U22 €R such thatcos a= u12- sin a+ u22cos asin a(iii) Use parts (i) and (ii) to show that B can be expressed in the formB = R,Ufor some a €R and some upper-triangular matrix U.(iv) Suppose that B = R,U = R3V, where a, B eR and U and V are upper-triangular. Prove that if B is invertible, then U = ±V.

This is a multipart question having two or more interrelated subparts (here we have four such…

[a b (e) Let B = be any 2 x 2 matrix. [cos a = u11 [sin a (i) Show that there are real numbers un and a such that Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for u in terms of a and c. (ii) Let a e R. Use the invertibility of R, to prove that there are unique U12, U22 € R such that -- sin a cos a [cos a u12 + u22 (sino [sin a] (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a €R and some upper-triangular matrix U. (iv) Suppose that B = RU = R3V, where a, B e R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V. %3D