4.3. EXERCISES 73
x=[1,2,3]; y=[2,3,4]; x.*y
and then press “enter”. You will get 2,6,12. You would get an error if you wrote x*y.Similarly, type
[2,4,6,8]./[1,2,3,4]
and press “enter”. This yields 2,2,2,2. The expression [2,4,6,8]/[1,2,3,4] doesn’t makeany sense.
4.3 Exercises1. Find the point (x1,y1) which lies on both lines, x+3y = 1 and 4x− y = 3.
2. Solve Problem 1 graphically. That is, graph each line and see where they intersect.
3. Find the point of intersection of the two lines 3x+ y = 3 and x+2y = 1.
4. Solve Problem 3 graphically. That is, graph each line and see where they intersect.
5. Do the three lines, x+2y = 1,2x− y = 1, and 4x+3y = 3 have a common point ofintersection? If so, find the point and if not, tell why they don’t have such a commonpoint of intersection.
6. Do the three planes, x+ y− 3z = 2, 2x+ y+ z = 1, and 3x+ 2y− 2z = 0 have acommon point of intersection? If so, find one and if not, tell why there is no suchpoint.
7. You have a system of k equations in two variables, k ≥ 2. Explain the geometricsignificance of
(a) No solution.
(b) A unique solution.
(c) An infinite number of solutions.
8. Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotesa nonzero number. Determine whether the given augmented matrix is consistent. Ifconsistent, is the solution unique?
■ ∗ ∗ ∗ ∗ | ∗0 ■ ∗ ∗ 0 | ∗0 0 ■ ∗ ∗ | ∗0 0 0 0 ■ | ∗
9. Here is an augmented matrix in which ∗ denotes an arbitrary number and ■ denotes
a nonzero number. Determine whether the given augmented matrix is consistent. Ifconsistent, is the solution unique? ■ ∗ ∗ | ∗
0 ■ ∗ | ∗0 0 ■ | ∗