# Two Matrix Solution and Explanation

SMITH, JOHNSON, AND COHEN LIVE IN BROOKLYN, MANHATTAN, AND THE BRONX (NOT NECESSARILY IN THAT ORDER). THEY ARE FLYING TO NEW YOUK CITY IN A JET WHOSE PILOT, COPILOT, AND NAVIGATOR ARE NAMED SMYTHE, JENSON, AND KOHN (AGAIN, NOT NECESSARILY RESPECTIVELY). IT IS KNOWN THAT

A. COHEN LIVES IN THE BRONX

B. JOHNSON IS DAF AND MUTE

C. SMYTHE HAD A BRIEF FLING WITH THE COPILOT'S WIFE

D. THE PASSENGER WHOSE NAME SOUNDS LIKE THE NAVIGATOR'S LIVES IN BROOKLYN. THE NAVIGATOR, HOWEVER, LIVES IN MANHATTAN.

E. THE NAVIGATOR'S NEXT-DOOR NEIGHBOR, ONE OF THE PASSENGERS, IS A FAMOUS OPERA SINGER.

WHAT ARE THE POSITIONS OF SMYTHE, JENSON, AND KOHN? (HINT: THERE ARE FOUR SETS OF INFORMATION. TWO MATRICES MIGHT BE USEFUL)

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#### Solution Summary

I show how to solve a logic problem where we are given limited information regarding people, their residences, and their occupations. Solving the problem requires drawing deductive inferences from the given information. The problem-solving process involves using two matrices.

Systems of Linear Equations : Row Operations and Solutions

1. Find the augmented matrix for each system of linear equations:

a. 5x1 + 7x2 + 8x3 = 3

-2x1 + 4x2 + 9x3 = 3

3x1 - 6x2 + x3 = 1

b. 4x1 + x2 - 7x3 = 6

5x1 + 7x2 + 2x3 = 3

5x1 + 2x2 + 5x3 = 7

c. 3x1 - 2x2 + 2x3 = 7

5x1 + 7x2 + 3x3 = 3

-5x1 + 6x2 - 8x3 = -5

2. Using elementary row operations reduce each of the augmented matrices from Problem 1 to reduced echelon form

3. Using the information from Problem 2, what are the solutions to the system of linear equations

4. Indicate whether the following statements are True or False

a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system. (Give a brief justification for your answer.)

b. Two matrices are row equivalent if they have the same number of rows. (Give a brief justification for your answer.)

c. An inconsistent system has more than one solution. (Give a brief explanation for your answer.)

d. Two linear systems are equivalent if they have the same solution set. (Give a brief explanation for your answer.)

5. Determine if the homogenous linear systems below have non-trivial solutions.

a. 2x1 - 5x2 + 8x3 = 0

-2x1 - 7x2 + x3 = 0

4x1 + 2x2 + 7x3 = 0

b. x1 - 3x2 + 7x3 = 0

-2x1 + x2 - 4x3 = 0

x1 + 2x2 + 9x3 = 0

Please see the attached file for the fully formatted problems.

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