Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000
Quiz 2 – Friday, September 10, 1999
NAME 0
1
0
1 2 ¡2
B
B
Let A = @ 2 0 1 C
A, B = @
¡1 1 1
1) The matrix B is a
£
Box
1
Ã
!
3 2
³
´
4 1 ¡2
1 4 C
,D= 3 1 4
A, C =
5 2 1
0 1
matrix.
2) Is it possible to multiply AB? Circle one: Yes or No. If Yes is your answer, what is the
size of AB?
3) Is it possible to multiply BA? Circle one: Yes or No. If Yes is your answer, what is the
size of BA?
4) Is it possible to multiply CD? Circle one: Yes or No. If Yes is your answer, what is the
size of CD?
5) Is it possible to multiply BC? Circle one: Yes or No. If Yes is your answer, what is the
size of BC?
6) Multiply the matrices B and D (you need to decide on the correct order).
7) Write the system of equations in matrix form; DON’T SOLVE.
2x ¡ 4y + 2z = 3
x ¡ 4y + 2z = 1
3x + 6y + 2z = 4
x ¡ y + 2z = 2
8) Apply the Gaussian Elimination process to the system 2x + y + 2z = 5 : Clearly show
3x + z
=4
the new systems (or augmented matrices) which the Elimination process produces and clearly
show the back substitution procedure. That is, be neat and orderly.
In 9-13, assume that A is a 3 £ 3 matrix and that you are solving the system of equations
¡
!
!
A¡
x = b:
9) In general, list the three possibilities for the number of solutions
.
0
1
0
1
0
1
0
1
¤ ¤ ¤ ¤
B
10) After applying the Gaussian Elimination process, the augmented matrix is @ 0 0 ¤ ¤ C
A
0 0 0 0
(a * means a non-zero number). How many free variables are there? Describe the solution
space.
¤ ¤ ¤ ¤
B
11) After applying the Gaussian Elimination process, the augmented matrix is @ 0 ¤ ¤ ¤ C
A
0 0 0 3
(a * means a non-zero number). How many free variables are there? Describe the solution
space.
¤ ¤ ¤ ¤
C
12) After applying the Gaussian Elimination process, the augmented matrix is B
@ 0 ¤ ¤ ¤ A
0 0 ¤ 0
(a * means a non-zero number). How many free variables are there? Describe the solution
space.
¤ ¤ ¤ ¤
C
13) After applying the Gaussian Elimination process, the augmented matrix is B
@ 0 0 0 0 A
0 0 0 0
(a * means a non-zero number). How many free variables are there? Describe the solution
space.
¡
!
!
14) Suppose that the output from Maple’s linsolve command when solving A¡
x = b is
¡21 ¡ _t1 + 5_t2
_t1
13 ¡ 3_t2
_t2
How many variables are in the system of equations?
How many columns does the matrix A have?
Is the system of equations homogeneous or non–homogeneous?
Describe the solution space.
The solutions are in what Rn space?
¡
!
Is it possible for b to be the zero vector?
Write down two di¤erent solutions to the system of equations.