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Dawson College - Mathematics Department
Final Examination - Linear Algebra 201-105-DW - Winter 2006
Marks
9% 1. Let
A B C
= ⎢^ ⎥ = ⎡^ ⎤^ =⎡^ ⎤
⎢ ⎥ ⎢^ ⎥^ ⎢^ −⎥
⎢ ⎥ ⎣^ ⎦^ ⎣^ ⎦
Compute each of the following, if possible: (a) 3 A − CT (b) AB (c) BA
6% 2. Compute the determinant: 0 1 1 1 2 4 4 0 1 0 3 2 1 2 5 1
7% 3. (a) Find A −^1 if
A
= ⎢^ ⎥
(b) Using the result in part (a) solve the system: 2 2 1 2 3 2 0 3 3 1
x y z x y z x y z
6% 4. Solve by the Gauss-Jordan elimination method: 1 2 3 1 2 3 1 2 3
x x x x x x x x x
8% 5. Let A and B be 3 × 3 matrices such that det A = 2 and B = − 3. Find
(a) det 2( A ) −^1 (b) det ( A BAT −^1 )
(c) det ( A^3 ) (d) det 3( B −^1 )
6% 6. Solve for x 3 only, using Cramer’s Rule: 1 2 3 1 2 3 1 2 3
x x x x x x x x x
18% 7. Given vectors u^ G^ = 2 i − j + 2 , k v G^ = i + j + k and w^ G^ = i − j + 3 k , Find the (a) cosine of the angle between u^ G^ and v^ G^. (b) area of the parallelogram with adjacent sides v^ G^ and w^ G^. (c) volume of the parallelepiped with adjacent edges u v^ G,^ G^ and w^ G^.
(d) equation of the line through point ( 1, 0, − 1 ) and perpendicular to the
vectors v^ G^ and w^ G^. (e) Pr oj vw G^ G^. (f) unit vector in the same direction as u^ G^.
10% 8. (a) Find the distance from the point P ( 3, −1, 4 ) to the
plane 2 x − y + 2 z − 10 = 0. (b) Find the equation of the plane containing the points
A (1, 0, 1 , ) B ( −2, 1, 0 ) and C ( 3, 1, − 1 ).
10% 9. (a) Find the distance from point P ( 4, − 2, 3) to the line
L : x = 1 − 2 , t y = 4 + t , z = − 3 + 2 t. (b) Find the point of intersection of line L from part (a) with the xz-plane.
10% 10. Use the Simplex Method to maximize P = 6 x 1 (^) + 4 x 2 (^) + 8 x 3 subject to 1 2 3 1 2 3 1 3
x x x x x x x x
x 1 (^) , x 2 (^) , x 3 ≥ 0.
- (a) 3 1 3 3 3
= 8. (a) (^53)
(b) ( 4, − 2, − 2 ) = 2 6 (b) x + 8 y + 5 z − 6 = 0
(c) 6
- (a) 9 (d) 1 1 4 2 2
x − (^) = y (^) = z + − −
(b) ( 9, 0, − 11 ) t = − 4
(e) 3 11
(f) 1 ( 2, 1, 2)
⎡ −^ − ⎤
( 0, 10, 10 ;)^ p^ =^120
( 10, 20, 0 ;) C = 5000
