Page 21 - Matrix Analysis & Applied Linear Algebra

1.2 Gaussian Elimination and Matrices 13

1.2.8. The following system has no solution:

−x 1 +3x 2 − 2x 3 =1,
−x 1 +4x 2 − 3x 3 =0,
−x 1 +5x 2 − 4x 3 =0.
Attempt to solve this system using Gaussian elimination and explain
what occurs to indicate that the system is impossible to solve.

1.2.9. Attempt to solve the system
−x 1 +3x 2 − 2x 3 =4,
−x 1 +4x 2 − 3x 3 =5,
−x 1 +5x 2 − 4x 3 =6,

using Gaussian elimination and explain why this system must have in-
finitely many solutions.

1.2.10. By solving a 3 × 3 system, find the coefficients in the equation of the
2
parabola y = α+βx+γx that passes through the points (1, 1), (2, 2),
and (3, 0).

1.2.11. Suppose that 100 insects are distributed in an enclosure consisting of
four chambers with passageways between them as shown below.

#3

#4 #2

#1

At the end of one minute, the insects have redistributed themselves.
Assume that a minute is not enough time for an insect to visit more than
one chamber and that at the end of a minute 40% of the insects in each
chamber have not left the chamber they occupied at the beginning of
the minute. The insects that leave a chamber disperse uniformly among
the chambers that are directly accessible from the one they initially
occupied—e.g., from #3, half move to #2 and half move to #4.