RENSSELAER POLYTECHNIC INSTITUTE

TROY, NY

RETEST NO. 1 INTRODUCTION TO ENGINEERING ANALYSIS

(ENGR-1100)

NAME: ______SECTION: ______

ID: ______October 1, 2003

INSTRUCTOR: ______
Problem / Points / Score
1 / 25
2 / 25
3 / 25
4 / 25
Total / 100

Problem 1 (25 Points)

1. Given the matrix A:

1 / 3 / -2
A= / 0 / -1 / 2
-3 / 1 / 0

(a) Find det(A), the determinant of A (5 points)

(b) Find the matrix of cofactors of A (10 points)

(c) Find adj(A), the adjoint of A (5 points)

(d) Find the inverse of A with adjoint formula, A-1 =[1/det(A)]*adj(A) (5 points)

NOTE: Show all work and calculations.

Problem 2 (25 Points)

Solve the following system of linear equations by Gaussian-Jordan Elimination (go all the way to row reduced echelon form).

x1 + x2 + x3 = 1

2x1 – 3x2 + 4x3 = 0

x1 + 5x2 - 3x3 = 5

NOTE: Show all work and calculations

Problem 3 (25 Points)

Solve the following system of linear equations for x1 by using Cramer’s rule. You must use the duplicate column method to evaluate the determinants.

x1 + 3x2 - 2x3 = 2

x1 + x2 = -3

- 2x2 + 5x3 = 2

NOTE: Show all work and calculations.

Problem 4 (25 Points)

Two points in 3D space have x, y, z coordinates as follows: P(2, 6, -1) and Q(4, 6, 2); a third point of interest is the origin of the coordinate system O(0, 0, 0).

Three vectors u, v and w are defined by joining these points as follows: u = OP, v = OQ, w = PQ.

Find:

(a) Angle q between u and v (5 points)

(b) Norm (length) of vector v (5 points)

(c) u x w (5 points)

(d) Vector of length = 1.0 (unit vector) parallel to w (10 points)

NOTE: Show all work and calculations.