Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
Quiz 1 - due Thursday (do in groups of 1, 2, or 3 persons)
1) Use Maple to solve the following problems. Be sure to put your answer in vector parametric
form and describe the solutions set (eg. no solution, a point, a line in R2 ; a line in R3 ; a
plane
in R3 ; a 4-space
in1R7 ; :::):
Hand
in one solution set0 per group. 1 0 1 0
0
10
0
1
1
3 1 2
x
0
3 1 2
x
¡1
CB
C
B
C
CB
C
B
C
a) B
b) B
@ 12 6 10 A @ y A = @ 0 A
@ 12 6 10 A @ y A = @ ¡6 A
¡9 1 ¡2
z
0
¡9 1 ¡2
z
¡1
0
1B
B
4
2 ¡1 6
2
B
CB
c)@ 16 10 ¡1 26 11 A B
B
B
¡12 ¡2 9 ¡11 1
@
0
0
1B
0
B
4
2 ¡1 6
2
CB
B
16
10
¡1
26
11
d)@
AB
B
B
¡12 ¡2 9 ¡11 1
@
x1 + 2x2 ¡ 3x3 + 3x4
3x1 + 7x2 ¡ 8x3 + 12x4
e)
2x1 + 5x2 ¡ 5x3 + 11x4
¡4x1 ¡ 6x2 + 14x3
x1
x2
x3
x4
x5
x1
x2
x3
x4
x5
1
0
1
C
C
0
C
C
C=B
@ 0 A
C
C
0
A
1
1
0
C
C
44
C
C
C=B
@ 181 A
C
C
¡112
A
x1 + 2x2 ¡ 3x3 + 3x4 = ¡20
3x1 + 7x2 ¡ 8x3 + 12x4 = ¡45
f)
2x1 + 5x2 ¡ 5x3 + 11x4 = ¡23
¡4x1 ¡ 6x2 + 14x3
= 116
=0
=0
=0
=0
2) Give an example of a 2x2 matrix A that has no inverse (you can easily create one by
remembering that a matrix has an inverse if and only if its determinant is non-zero). What
does Maple give you if you use the command
! by hand, that A has no
!
à Prove,
à inverse(A)?
1 0
a b
. Now obtain a contradic=
inverse by setting up the matrix equation A
0 1
c d
tion multiplying the matrices on the left and setting the product equal to the identity on
the right.
3) For the matrix A that you used in (7), …nd (and sketch in R2 ) the null space of A: You
will need to search through the notes to …nd what null space means.
4) Given examples of 2x2 matrices A and B such that AB 6= BA:
0
10
1
0
1
1 ¡2 3
x
1
B
CB
C
B
5) The equation @ 2
3 1 A@ y A = @ 7 C
A has a unique solution. Verify this state¡5 2 1
z
¡2
ment in 3 di¤erent ways: (a) use linsolve, (b) use solve, (c) use the inverse of the matrix.