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This is the Exam of Linear Algebra and its key important points are: Simplify, Values, Compute, Matrices, Commute Under Multiplication, Equations, Reduction Method, Particular Solution, Inconsistent, Non Trivial Solutions
Typology: Exams
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6% 1. a) For what values of x will
x x
b) Compute
T −^ ^ (^) − (^) − (^) (^) −
6% 2. Simplify
1 1 7 A B − −
6% 3. a) Find A −^1 if A^7^ − 5 A^4^ + 3 A^2 + 2 I = 0
b) Prove: If A and B are matrices which commute under multiplication, then (^) AT and (^) BT also commute under multiplication. 5% 4. a) Solve the following system of equations using the Gauss-Jordon reduction method.
1 2 1 2 3 1 2 3
x x x x x x x x
b) What is the particular solution when x 3 (^) = a = − 4.
6% 5. a) Find the condition on k 1 (^) , k 2 and k 3 so that the system will be inconsistent: 1 2 3
x y z k x y z k x y k
d) unit vector in the opposite direction to C
ur .
e) Pr oj CA ur
ur .
9% 11. a) Find the point of intersection of the line (^) x = 4 − 2 , t y = 2 + 2 , t z = 3 − 4 t with the yz plane.
b) Find an equation of the line through point P (^) ( 3, −1, 2 (^) )and perpendicular to the plane through points Q (^) ( −1,1, 2 , (^) ) R ( 1, −2,1) and S ( 2, 2, 4 .)
c) Find the distance from point (^) ( 4, −2,3 (^) ) to the line x = 1 − 2 , t y = 4 + t , z = − 3 + 2 t.
9% 12. a) Find an equation of the plane containing the points
P (^) ( 1, 0,1 ,) Q (^) ( −1, −4,1 (^) ) and R (^) ( −2, −2, 2 (^) ).
b) Find the distance from point (^) ( 3, −4,1 (^) )to the plane x − 2 y + 2 z + 4 = 0. c) Find the line of intersection of the planes x + y + 2 z − 6 = 0 and 2 x + y − z − 4 = 0.
10% 13. Maximize Z = x 1 (^) + 2 x 2 (^) + x 3 (^) + 5 x 4
Subject to x 1 (^) + x 3 (^) + x 4 ≤ 50 1 2 3 4 1 2 3 4
x x x x x x x x
10% 14. Write the dual problem and use it to:
Minimize C = 200 x 1 (^) + 150 x 2 (^) + 400 x 3
Subject to 2 x 1 (^) + x 2 (^) + 3 x 3 ≥ 20 1 2 3 1 3 1 2 3
x x x x x x x x
