Matrix - Assignment

Pure Mathematics – Matrices and Determinants

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Matrices and Determinants Assignment 1. (a) Let R be the matrix representation the rotation in the Cartesian plane anticlockwise about the origin by 60o. (i) Write down R and R6. ⎛2 1 ⎞ ⎟⎟ . Verify that A-1RA is a matrix which all the elements (ii) Let A = ⎜⎜ 3⎠ ⎝0 are integers. (b) Using the results of (a), or otherwise, find a 2x2 matrix M, in which all the ⎛1 0⎞ ⎟⎟ . elements are integers, such that M3 = 1 but M ≠ 1, where I = ⎜⎜ ⎝0 1⎠ 2.

Let T1 be the transformation which transform a vector x to a vector y = Ax, where ⎛ ⎜ A=⎜ ⎜ ⎜ ⎝

3 2 1 2

1⎞ − ⎟ 2⎟. 3⎟ ⎟ 2 ⎠

⎛0⎞ Find y when x = ⎜⎜ ⎟⎟ ⎝ 2⎠ (ii) Describe the geometric meaning of the transformation T1. (iii) Find A2005. (b) For every integer n greater than 1, let Tn be the transformation which transform a vector x to a vector y = Anx. (a) (i)

⎛0⎞ Is there a positive integer m such that the transformation Tn transform ⎜⎜ ⎟⎟ ⎝ 2⎠

⎛− 2 ⎞ ⎟ ? Explain your answer. to ⎜⎜ ⎟ 2 ⎝ ⎠

Pure Mathematics – Matrices and Determinants

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Consider the system of linear equations in x, y and z. ⎧ x + ay + z = 4 (E ) : ⎪⎨ x + (2 − a ) y + (3b − 1)z = 3 , where a, b ∈ R. ⎪2 x + (a + 1) y + (b + 1)z = 7 ⎩

(a) Prove that (E) has a unique solution if and only if a ≠ 1 and b ≠ 0. Solve (E) in this case. (b) (i) For a = 1, find the value(s) of b for which (E) is consistent, and solve (E) for such value(s) of b. (ii) Is there a real solution (x, y, z) of ⎧x + y + z = 4 ⎪ ⎨2 x + 2 y + z = 6 ⎪4 x + 4 y + 3 z = 14 ⎩

Satisfying x2 – 2y2 – z = 14? Explain your answer. (c) Is (E) consistent for b = 0? Explain your answer.

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⎛ m − m⎞ ⎟⎟ , where m > 0. Let M = ⎜⎜ ⎝m m ⎠ (a) Evaluate M2. ⎛a b ⎞ ⎟⎟ be a non-zero real matrix such that MX = XM. (b) Let X = ⎜⎜ ⎝c d ⎠ (i) Prove that c = -b and d = a. (ii) Prove that X is a non-singular matrix. ⎛1 0⎞ ⎟⎟ . (iii) Suppose that X − 6 X −1 = ⎜⎜ ⎝0 1⎠ (1) Find X. (2) If a > 0 and (M – kX)2 = –M2, express k in terms of m. (c) Using the result of (b)(iii)(2), find two real matrices P and Q, other than M and –M, such that P4 = Q4 = M4.