Engr231exam1prepsolution

# Engr231exam1prepsolution - ENGR 231 Linear Engineering...

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ENGR 231 Exam 1 Practice Problems Solutions 1 ENGR 231 Linear Engineering Practice Problems for Exam I Solution 1. Find the general solution of the following system of equations. Write your answer in parametric vector form. x 1 ! 5 x 2 ! 9 x 3 + 8 x 4 = ! 7 x 2 + 3 x 3 ! 4 x 4 = 2 2 x 2 + 6 x 3 ! 8 x 4 = 4 Answer : Augmented matrix and RREF 1 ! 5 ! 9 8 ! 7 0 1 3 ! 4 2 0 2 6 ! 8 4 " # \$ \$ \$ % ' ' ' 1 0 6 ! 12 3 0 1 3 ! 4 2 0 0 0 0 0 " # \$ \$ \$ % ' ' ' . Basic variables: 1, 2. Free variables 3, 4. x 1 + 6 x 3 ! 12 x 4 = 3 x 2 + 3 x 3 ! 4 x 4 = 2 x 3 free x 4 free x 1 x 2 x 3 x 4 " # \$ \$ \$ \$ \$ % ' ' ' ' ' = 3 ! 6 x 3 + 12 x 4 2 ! 3 x 3 + 4 x 4 x 3 x 4 " # \$ \$ \$ \$ \$ % ' ' ' ' ' = 3 2 0 0 " # \$ \$ \$ \$ % ' ' ' ' + x 3 ! 6 ! 3 1 0 " # \$ \$ \$ \$ % ' ' ' ' + x 4 12 4 0 1 " # \$ \$ \$ \$ % ' ' ' ' 2. Find the product of the matrix A and the vector x = (2, -1). A = ! 4 12 1 ! 3 ! 3 8 " # \$ \$ \$ % ' ' ' Answer: ! 4 12 1 ! 3 ! 3 8 " # \$ \$ \$ % ' ' ' 2 ! 1 " # \$ % ' = 2 ! 4 1 ! 3 " # \$ \$ \$ % ' ' ' + ( ! 1) 12 ! 3 8 " # \$ \$ \$ % ' ' ' = ! 8 ! 12 2 + 3 ! 6 ! 8 " # \$ \$ \$ % ' ' ' = ! 20 5 ! 14 " # \$ \$ \$ % ' ' ' 3. Are the following matrices in echelon form? Are they in the reduced echelon form? Justify your answer. ( a ) 1 2 3 4 0 2 8 2 0 0 1 7 0 0 0 0 ! " # # # # \$ % ( b ) 1 ' 8 0 4 0 0 1 ' 2 0 0 0 0 0 0 0 0 ! " # # # # \$ % Answer : (a) Echelon form: values of 0 below and to the left of each pivot. (b) Reduced echelon form. 1's on pivot positions, 0 above and below each pivot,

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ENGR 231 Exam 1 Practice Problems Solutions 2 4. The arrays in Problem 3 are augmented matrices for linear systems. For each matrix, answer the following questions: (a) Identify the free and basic variables. (b) State whether the systems are consistent. (c) For consistent systems, find a particular solution. (d) For systems with a free variable, find a nontrivial solution to the homogeneous equation. (e) Verify that the sum of a particular and homogenous solutions is also a solution of the linear system. (f)
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## Engr231exam1prepsolution - ENGR 231 Linear Engineering...

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