(1) [1.1 #8 p. 11] For A =  [2 -1 5 3 4 1] , B =  [1 0 -2 2 3 4]  , compute (A + 2B) T .(2) Write the 4 × 4 matrix A = (aij )4×4 whose (i, j) entry is aij = 2i - 3j.(3) [1.1 #33 p. 11] x East y North FF 30? An airplane is flying with a ground speed of 300 mph at an angle of 30? east of due north. In addition, the airplane is climbing at a rate of 10 mph. Determine the vector in R 3 that represents the velocity (in mph) of the airplane.(4) [1.1 #82 p. 12] The trace of an n × n matrix A = (aij ), written tr(A), is defined to be the sum of the diagonal elements: tr(A) = a11 + a22 + · · · + ann. Prove that, for any n × n matrices A and B and scalar c, the following statements are true: (a) tr(A + B) = tr(A) + tr(B). (b) tr(cA) = c · tr(A). (c) tr(AT ) = tr(A).(5) [1.2 #9 p. 25] Compute ? ? s( 0 0 0) t( 0 0 0) u ? ? ? ? a b c ? ?.(6) [1.2 #35 p. 25] Determine whether u ? span{v1, v2} in R 2 , where u = (-1, 11), v1 = (1, 3), v2 = (2, -1). If it does, write u as a linear combination of v1 and v2. (7) [1.2 #83 p. 26, Theorem 1.3 (b)] Prove that if A is an m × n matrix, u = (x1, x2, . . . , xn) T is a column vector, and c ? R, then c(Au) = (cA)u = A(cu). (When u might have more than one column, see 2.1 #54 p. 105 on a subsequent homework.)(8) [1.3 #1 p. 38] Write (a) the coefficient matrix and (b) the augmented matrix of the following system: -x2 + 2x3 = 0, x1 + 3x2 = -1.(9) [1.3 #55 p. 39] Suppose that the general solution of a system of m linear equations in n variables contains k free variables. How many basic variables does it have?(10) [1.3 #56 p. 39] Suppose that R is a matrix in reduced row echelon form. If row 4 of R is nonzero and has its leading entry in column 5, describe column 5.(11) [1.4 #6 p. 52] Determine whether the following system is consistent. If it is, write the answer in vector form: x1 - 2x2 - x3 = 3, -2x1 + 4x2 + 2x3 = -6, 3x1 - 6x2 - 3x3 = 9. (12) [1.4 #24 p. 52] For which r, if any, is the system x1 + rx2 = 2, rx1 + 16x2 = 8 inconsistent?