1. (a) Let A be a 2 X 2 matrix with detA = 3. Compute the determinant of the matrix A 1 2 B _ 3 4 _ 0 0 0 5 O 0 6 0 (b) Let C be a 3 x 3 matrix which is not invertible and let D be the 4 X 4 matrix 0 C 0 D _ 0 'r s t 1 where r, s, and t are real numbers. Explain Why D is not invertible. Q2. Find a basis for Col A and Nul A for the following matrix 1 O 2 0 1 1 -3 1 1 4 0 1 O 1 1 0 2 1 0 O 0 1 O 2 0 1 O 1 Q3. Let C be a 2 x 3 matrix such that Cx = b has a solution for every b E R2 and let D be a 3 X 2 matrix such that Dx = 0 has only the trivial solution x = 0. a) Explain Why the product DC is never invertible. b) Is the product CD always invertible? Q4. Let P2 denote the space of all polynomials of degree less than or equal to 2. a) Does the set 8 = {(t1)(t 2),(t+1)(t+ 2),t} form a basis for P2? b) The set 15' = {t2 + t + 1,132 + 275+ 1,315+ 1} forms a basis for P2. Find the coordinates of (30$) 2 3152 + t 1 in the 8 basis. Q5 a) Let V denote the vector Space of all 2 by 2 matrices. Is the map T : V > R given by T(A) = det A a linear transformation? b) Let V be the set of all continuous functions f where f (0) < f (1). Is V a vector Space? 0) Let V be the set of all odd functions, that is f(:L') = f(:r:) for every real number x. Is V a vector space? Q6. Consider the Sphere of radius 1 centered at the origin in R3, whose volume is gar. We transform the sphere via a linear transformation whose matrix in the standard basis is given by 3 2 1 A: 1 1 1 1 1 1 What is the volume of the resulting shape? What is det A9 for the above A