1. Let 2 4 1 3 2 5 A , B , C 3 2 2 5 3 4 = = = Find each of the following: (i) A + B (ii) A B (iii) 3A C (iv) AB (v) BA 2. Compute the following: (i) a b a b b a b a + (ii) 2 2 2 2 2 2 2 2 2 2 2 2 a b b c ab bc a c a b ac ab + + + + + (iii) 1 4 6 12 7 6 8 5 16 8 0 5 2 8 5 3 2 4 + (iv) 2 2 2 2 2 2 2 2 cos sin sin cos sin cos cos sin x x x x x x x x + 3. Compute the indicated products. (i) a b a b b a b a (ii) 1 2 3 [2 3 4] (iii) 1 2 1 2 3 2 3 2 3 1 (iv) 2 3 4 1 3 5 3 4 5 0 2 4 4 5 6 3 0 5 (v) 2 1 1 0 1 3 2 1 2 1 1 1 (vi) 2 3 3 1 3 1 0 1 0 2 3 1 4. If 1 2 3 3 1 2 4 1 2 A 5 0 2 , B 4 2 5 and C 0 3 2 1 1 1 2 0 3 1 2 3 = =
Question 1:Let 2 4 1 3 2 5 A , B , C 3 2 2 5 3 4 − = = = − Find each of the following: (i) A + B (ii) A – B (iii) 3A – C (iv) AB (v) BA
Question 2:Compute the following: (i) a b a b b a b a + − (ii) 2 2 2 2 2 2 2 2 2 2 2 2 a b b c ab bc a c a b ac ab + + + + + − − (iii) 1 4 6 12 7 6 8 5 16 8 0 5 2 8 5 3 2 4 − − + (iv) 2 2 2 2 2 2 2 2 cos sin sin cos sin cos cos sin x x x x x x x x +
Question 3:Compute the indicated products. (i) a b a b b a b a − − (ii) 1 2 3 [2 3 4] (iii) 1 2 1 2 3 2 3 2 3 1 − (iv) 2 3 4 1 3 5 3 4 5 0 2 4 4 5 6 3 0 5 − (v) 2 1 1 0 1 3 2 1 2 1 1 1 − − (vi) 2 3 3 1 3 1 0 1 0 2 3 1 − − −
Question 4:If 1 2 3 3 1 2 4 1 2 A 5 0 2 , B 4 2 5 and C 0 3 2 1 1 1 2 0 3 1 2 3 − − = = = − − , then compute (A+B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.
Question 5:If 2 5 2 3 1 1 3 3 5 5 1 2 4 1 2 4 A and B 3 3 3 5 5 5 7 2 7 6 2 2 3 3 5 5 5 = = , then compute 3A – 5B.
Question 6:Simplify cos sin sin cos cos + sin sin cos cos sin q q q − q q q − q q q q
Question 7:Find X and Y, if (i) 7 0 3 0 X + Y and X – Y 2 5 0 3 = = (ii) 2 3 2 2 2X + 3Y and 3X 2Y 4 0 1 5 − = + = −
Question 8:Find X, if Y = 3 2 1 4 and 2X + Y = 1 0 3 2 −
Question 9:Find x and y, if 1 3 0 5 6 2 0 1 2 1 8 y x + =
Question 10:Solve the equation for x, y, z and t, if 1 1 3 5 2 3 3 0 2 4 6 x z y t − + =
Question 11:If 2 1 10 3 1 5 x y − + = , find the values of x and y.
Question 12:Given 6 4 3 1 2 3 x y x x y z w w z w + = + − + , find the values of x, y, z and w.
Question 13:If cos sin 0 F ( ) sin cos 0 0 0 1 x x x x x − = , show that F(x) F(y) = F(x + y).
Question 14:Show that (i) 5 1 2 1 2 1 5 1 6 7 3 4 3 4 6 7 − − ¹ (ii) 1 2 3 1 1 0 1 1 0 1 2 3 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 2 3 4 2 3 4 1 1 0 − − − ¹ −
Question 15:Find A2 – 5A + 6I, if 2 0 1 A 2 1 3 1 1 0 = −
Question 16:If 1 0 2 A 0 2 1 2 0 3 = , prove that A3 – 6A2 + 7A + 2I = 0
Question 17:If 3 2 1 0 A and I= 4 2 0 1 − = − , find k so that A2 = kA – 2I
Question 18:If 0 tan 2 A tan 0 2 a − = a and I is the identity matrix of order 2, show that I + A = (I – A) cos sin sin cos a − a a a
Question 19:A trust fund has ` 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide ` 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) ` 1800 (b) ` 2000
Question 20:The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are ` 80, ` 60 and ` 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra. Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and π × k, respectively. Choose the correct answer in Exercises 21 and 22
Question 21:The restriction on n, k and π so that PY + WY will be defined are: (A) k = 3, π = n (B) k is arbitrary, π = 2 (C) π is arbitrary, k = 3 (D) k = 2, π = 3
Question 22:If n = p, then the order of the matrix 7X – 5Z is: (A) π × 2 (B) 2 × n (C) n × 3 (D) π × n
