16 6. (i) A = −     2 4 a a b where a and b are non-zero constants. Given that the matrix A is self-inverse, (a) determine the value of b and the possible values for a. (5) The matrix A represents a linear transformation M. Using the smaller value of a from part (a), (b) show that the invariant points of the linear transformation M form a line, stating the equation of this line. (3) (ii) P = −     p p p 2 1 3 where p is a positive constant. The matrix P represents a linear transformation U. The triangle T has vertices at the points with coordinates (1, 2), (3, 2) and (2, 5). The area of the image of T under the linear transformation U is 15 (a) Determine the value of p. (4) The transformation V consists of a stretch scale factor 3 parallel to the x‑axis with the y-axis invariant followed by a stretch scale factor –2 parallel to the y‑axis with the x-axis invariant. The transformation V is represented by the matrix Q. (b) Write down the matrix Q. (2) Given that U followed by V is the transformation W, which is represented by the matrix R, (c) find the matrix R. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________