Math 3013
SAMPLE FIRST EXAM
(with solutions)
1. (10 pts) Show that the points (1,0,1,1) , (1,1,0,1), and (1,2,3,1) form the vertices of a right triangle in
R4.
•If A= (1,0,1,1) ,B= (1,1,0,1), and C= (1,2,3,1))are the vertices of a triangle, then the sides
of the triangle correspond to the vectors
AB =B−A= (0,1,−1,0)
AC =C−A= (0,2,2,0)
BC =C−B= (0,1,3,0)
Since
AB ·AC = (0)(0) + (1)(2) + (−1)(2) + (0)(0) = 0 + 2 −2 + 0 = 0
the first two sides must be perpendicular. Hence the points A,Band Cform the vertices of a
right triangle in R4
2. Consider the following linear system
x1−x2+ 2x3= 1
2x1+x2+x3= 5
−x1+x2+ 2x3=−1
x2+x3= 1
(a) (5 pts) Write down the corresponding augmented matrix and reduce it to row-echelon form.
1−1 2
2 1 1
−112
0 1 1
1
5
−1
1
R2→R2−2R1
R3→R3+R1
−−−−−−−−−−−−−→
1−1 2
0 3 −3
0 0 4
0 1 1
1
3
0
1
R4→R4−1
3R2
−−−−−−−−−−−−→
1−1 2
0 3 −3
0 0 4
0 0 2
1
3
0
0
R4→R4−1
2R2
−−−−−−−−−−−−→
1−1 2
0 3 −3
0 0 4
0 0 0
1
3
0
0
(b) (5 pts) Reduce the augmented matrix further to reduced row-echelon form.
R2→1
3R2
R3→1
4R3
−−−−−−−−−→
1−1 2
0 1 −1
0 0 1
0 0 0
1
1
0
0
R2→R2+R3
R1→R1+R2
−−−−−−−−−−−−→
101
010
001
000
2
1
0
0
R1→R1−R3
−−−−−−−−−−→
100
010
001
000
2
1
0
0
(c) (5 pts) Write down the solution of the original linear system.
•x1= 2 , x2= 1 , x3= 0
1
