Page 63 - Matrix Analysis & Applied Linear Algebra

56 Chapter 2 Rectangular Systems and Echelon Forms

2.3.4. Consider two consistent systems whose augmented matrices are of the
form [A|b] and [A|c]. That is, they differ only on the right-hand side.
Is the system associated with [A | b + c] also consistent? Explain why.

2.3.5. Is it possible for a parabola whose equation has the form y = α+βx+γx 2
to pass through the four points (0, 1), (1, 3), (2, 15), and (3, 37)? Why?

2.3.6. Consider using floating-point arithmetic (without scaling) to solve the
following system:
.835x + .667y = .168,
.333x + .266y = .067.

(a) Is the system consistent when 5-digit arithmetic is used?
(b) What happens when 6-digit arithmetic is used?

2.3.7. In order to grow a certain crop, it is recommended that each square foot
of ground be treated with 10 units of phosphorous, 9 units of potassium,
and 19 units of nitrogen. Suppose that there are three brands of fertilizer
on the market— say brand X , brand Y , and brand Z . One pound of
brand X contains 2 units of phosphorous, 3 units of potassium, and 5
units of nitrogen. One pound of brand Y contains 1 unit of phosphorous,
3 units of potassium, and 4 units of nitrogen. One pound of brand Z
contains only 1 unit of phosphorous and 1 unit of nitrogen. Determine
whether or not it is possible to meet exactly the recommendation by
applying some combination of the three brands of fertilizer.

2.3.8. Suppose that an augmented matrix [A|b] is reduced by means of Gaus-
sian elimination to a row echelon form [E|c]. If a row of the form

(0 0 ··· 0 | α ) , α
=0

does not appear in [E|c], is it possible that rows of this form could have
appeared at earlier stages in the reduction process? Why?