4 t 1 - 2 2 0 1 t 1. Let A = where t is a real number. " 3 03 -1 -1 -t 1 2 (a) Evaluate det (A). (b) Find all values of t for which the matrix A is invertible. (c) Read about the adjugate method for calculating A-1 on p. 130-134 of the textbook. Using the formula for A- involving adj (A) (on p. 131) calculate the (3,4)-th entry of A when t1. (d) Using the results obtained in parts (a) and (c) find each of the following, for t = 1 : (i) det (-A). (ii) det (45). (iii) det (AAT). (iv) det (3A-1). (v) det(adj (A)). 2. Consider the following three vectors 2 0 : = : = 3 81 -3 and w= 1 (a) Show that u, v and ware not all coplanar (i.e. that they do not all lie in the same plane). (b) Calculate the cosine of the angle between 7 and the line segment from A(-1,4,1) to B(3, 1, 2). Express your answer in exact form, but estimate the value of the angle itself. (c) Calculate the projection of -37 onto w and the component of -37 that is orthogonal to w. Express your answers in exact form, and explicitly check that both components are orthogonal to each other. (d) Evaluate (3) w and check that the result is orthogonal to both - 3 and w. (e) Explain what is wrong with the expressions: (i) (. v) + w. (ii) uxvxw.