Exercise 4.5

Prove that the determinant sin cos - sin - 1 cos 1 x x x θ θ θ θ is independent of θ. 2. Evaluate cos cos cos sin - sin - sin cos 0 sin cos sin sin cos α β α β α β β α β α β α . 3. If A-1 = ( ) 3 1 1 1 2 2 15 6 5 and B 1 3 0 , find AB 5 2 2 0 2 1 - - - - - - - -         =           4. Let A = 1 2 1 2 3 1 1 1 5             . Verify that (i) [adj A]-1 = adj (A -1) (ii) (A-1)-1 = A

x + y + z = 1 5. 3x-y - 2z = 2 6. 5x - y + 4z = 5 2x + 3y + 2z = 2 2y - z = -1 2x + 3y + 5z = 2 ax + ay + 2az = 4 3x - 5y = 3 5x - 2y + 6 z = -1 Solve system of linear equations, using matrix method, in Exercises 7 to 14. 7. 5x + 2y = 4 8. 2x - y = -2 9. 4x - 3y = 3 7x + 3y = 5 3x + 4y = 3 3x - 5y = 7 10. 5x + 2y = 3 11. 2x + y + z = 1 12. x - y + z = 4 3x + 2y = 5 x - 2y - z = 3 2 2x + y - 3z = 0 3y - 5z = 9 x + y + z = 2 13. 2x + 3y +3 z = 5 14. x - y + 2z = 7 x - 2y + z = - 4 3x + 4y - 5z = - 5 3x - y - 2z = 3 2x - y + 3z = 12 15. If A =

3 2 - 4 / 1 1 -2 /   /        , find A-1. Using A-1 solve the system of equations 2x - 3y + 5z = 11 3x + 2y - 4z = - 5 x + y - 2z = - 3 16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is ` 70. Find cost of each item per kg by matrix method.

Evaluate

• / + / +

Evaluate 1/1 x y

x x+ y + Using properties of determinants in Exercises 11 to 15, prove that:

Solve the system of equations 2 3 10 4+ + =x y z 4 6 5 1+ =-x y z 6 9 20 2+ = -x y z Choose the correct answer in Exercise 17 to 19.

0 0 / x / y / z    =      is (A) 1/1 / 0 0 / 0 0 / 0 0 / x / y / z − − −           (B) 1/1 / 0 0 / 0 0 / 0 0 / x xyz y z − − −           (C) 0 0 / 1 0 0 / 0 0 / x yxyz z          (D) 1 0 0 / 1 0 1 0 0 0 1xyz         

Let A = 1 sin 1 sin 1 sin 1 sin 1 θ   − θ θ   − − θ  , where 0 ≤ θ ≤ 2π. Then (A) Det (A) = 0 (B) Det (A) ∈ (2, ∞) (C) Det (A) ∈ (2, 4) (D) Det (A) ∈ [2, 4]