Theory question 112
Matrices and Determinants(a) The point P(3, −5) is rotated through an angle 60° anticlockwise about the origin. (i) Obtain the matrix for the rotation. (ii) Find the image P′ of point P under the rotation.
(b) A linear transformation is given by N: (x, y) → (2x + 3y, 3x − y). (i) Write down the matrix N of the transformation. (ii) If N2 + aN + bI = O, where a, b ∈ R, I is the 2 × 2 unit matrix and O is the 2 × 2 null matrix, find the values of a and b.
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(a)(i) (cos 60° −sin 60°; sin 60° cos 60°) = (1/2 −√3/2; √3/2 1/2)
(ii) x′ = 3/2 + 5√3/2 = (3 + 5√3)/2, y′ = 3√3/2 − 5/2 = (3√3 − 5)/2
P′((3 + 5√3)/2, (3√3 − 5)/2) ≈ (5.83, 0.10)
(b)(i) N = (2 3; 3 −1)
(ii) N2 = (13 3; 3 10)
13 + 2a + b = 0 and 3 + 3a = 0
a = −1, b = −11 (check: 10 − a·1 + b = 10 + 1 − 11 = 0)
Matrices are written (row 1; row 2). Compare corresponding entries of N² + aN + bI with zero.