Download Linear Algebra Exam: Solutions for Gauss-Jordan, Inverse Matrices, Determinants, Vectors and more Exams Linear Algebra in PDF only on Docsity!
DAWSON COLLEGE
MATHEMATICS DEPARTMENT
FINAL EXAMINATION
LINEAR ALGEBRA
201-NYC-05 (SCIENCE)
MAY 2005 9:30-12:
- Given the linear system. 1 2 3 4 1 2 3 4 1 2 3 4
x x x x x x x x x x x x
(a) Find the general solution by using Gauss-Jordan elimination. (b) Find the particular solution for which x 1 = 7.
- Given 2 5 1 3 5 13 1 2 3
x y z x y z x y z
(a) Find the inverse of the coefficient matrix. 1 2 5 3 5 13 2 1 1
A
⎡ −^ − ⎤
= ⎢^ − − ⎥
(b) Use the result in (a) to find the solution of the given system of equations.
- (a) If (^) ( ) 1
I A −^
⎡ −^ − ⎤
find A. (b) Given the matrices. 2 1 (^1 2) , 1 1 3 , 0 1 3 5 2 3 1 1 4
A B C
−^ ⎡^ ⎤
= ⎡^ ⎤^ = ⎡^ ⎤^ =⎢^ ⎥
,
find (^) A −^1 B − 3 CT.
2
- (a) Given the matrix
a b c A d e f g h i
= ⎢^ ⎥
with det (^) ( A ) = − 2.
find ( ) ( ) ( ) ( ) ( ) ( )
g a i c h b d g f i e h a c b
.
(b) Use Cramer’s rule to solve for z. 6 2 2 5 2 1
x y z x y z x y z
- (a) Given a square matrix A. Prove that if P −^1 AP is invertible for some matrix P, then A is invertible. (b) Given the point A^ ( 1, 2,^ −^5 ) and the plane PL : x + 3 y + 2 z − 11 = 0. (i) Find the parametric equations of the line passing through the point A and perpendicular to the plane PL. (ii) Find the point on the plane PL closest to the point A.
- Given the vectors u = (^) ( 1, 2, 1) = i + 2 j + k , v = (^) ( 2, 1, 1) = 2 i + j + k
K K^ K^ K^ K K^ K K
. Find: (a) the cosine of the angle between u^ K^ and v^ K^. (b) Pr ojv K^ ( u K^ ). (c) a unit vector in the same direction as v^ K^.
- Given lines L : 1 x = − 2 + t , y = 13 − t , z = − 14 − 2 t , L 2 : x = − + 1 3 , s y = 2 + 4 , s z = − − 1 s , L : x = 14 + 9 , r y = 5 − 5 , r z = 4 + 7 r. (a) Show that L is perpendicular to L 1 and L 2. (b) Find the point Q of intersection of L with L 2. (c) Use the result of (b) and the fact that P ( −4, 15, − 10 )is the point of intersection of L with L 1 to find the (shortest) distance between the lines L 1 and L 2.
