Transcribed Text
1. (1pt) If
5.(1 pt)
True False Problem
4
4 2 0
A
3
and
2
4
1
Enter Tor F depending on whether the statement is true or
4 2
2 1
false. (You must enter Tor F True and False will not work.)
then
1. If matrices A and B have the same dimension, then
3(A+B)=3A+3B.
AB

2. The ith row, jth column entry of a square matrix, where
j. is called a diagonal entry.
and
3. The 3rd row, 4th column entry of matrix is below and
to the right of the 2nd row, 3rd column entry.
BA
6. pt) If
5
6
2. pt) If
A
6
6
2
3
A
and B =
10 6
10
6
2
1
6
7
then
B
7
2
6
9
A(2B)
5
6
5
1
then the dimension of AB is
and the dimension of BA is
3. (1 pt) Determine the value(s) of such that
4
5
4
NOTE: If either of the products is not defined, type UNDE
x
[x21]
5
1
=
FINED for you answer. If the product is defined, type the
4
2
dimension in the form mxn with NO spaces in betwee.
7. (1 pt)
If A and B are 2x 4 matrices, and C is a 9x 2 matrix, which of
the following are defined?
Note: If there is more than one value write them separated
by commas.
A. BC
B. B2
4. (1 pt) Perform the following operation:
B2 or
B
a
a
4 4
5
B
1+a
a
4
2
CA
8. (1 pt) If A, B. and C are 4 4. 4x' 7. and 7x 6 matri
ces respectively, determine which of the following products are
defined. For those defined, enter the dimension of the resulting
matrix (e.g. "3 4", with spaces between numbers and "x").
For those undefined, enter "undefined".
CB:
Note: The entries in the resulting matrix are functions of a.
AB:
AC:
A²:
in matrix form.
x
9. (1 pt) Compute the following product.
y
=

1
2
;
=
1

14. (1 pt) Find a and b such that
34
9
10. (1 pt) Compute the following product.
2
1
tb
1
24
2
7
=
]
b =
4
15. (1 pt) Write a vector equation
11.(1 Find x3 matrix A such that

1
x+

y+
2=

A
1
3
2
that is equivalent to the system of equations:
A
1
1
and
6y
Z
8
5
7x + 3y + Z 3
5
2x
+
y
3z
=
1
A
3
1
3
4
5
8
A
then un
=
12. (1 pt) Compute the following products.
2
2
5
10
20
55
2
=
17. pt) Let A
5 9 19
and b =
51
5
24
2
2
2
2
=
1. Determine if b is linear combination of a1, az and a3,
the columns of the matrix A.
13. (1 pt) Write the system
If it is a linear combination, determine a nontrivial linear rela
2x+9==5
tion (a nontrivial relation is three numbers which are not all
4x2y =3
three zero.) Otherwise, enter O's for the coefficients.
Grt3y7z=10
a
82
a3 b.
1. (1 pt) Given the matrix A
find A³.
5.(1pt)IfA=
,
determine the values of x and y for
which A² = A.
2 1
y
2.(1 pt) If A =
373
1
and B =
1 0 4
6. (1 pt) Find a nonzero, twobytwo matrix such that:
2
0 4 0
3
9

6
18
*
=


:of
TO

Then 4A B
7. (1 pt) Find a nonzero, twobytwo matrix such that:


=

AT
8. (1 pt) Let A be a 3x2 matrix. Suppose we know that
2
3
u
and
y
satisfy the equations Au = a and
BT
5
1
Av b. Find a solution x to Ax = 5a 2b.

x
B7 AT
9. (1 pt) Let A and B be symmetric n x n matrices. For each
of the following, determine whether the given matrix must be
(AB)T
symmetric or could be non symmetric.
1. E AB
6
3. =
22]
2. G = AB BA
Find a and C2 such that M² = 0, where I2 is the
3. H = AB BA
identity 2 2 matrix and 0 is the zero matrix of appropriate di
4. F ABA
mension.
5. C A + B
C1
6. D A2
c2
4. (1 pt)
10. (1 pt) Find the inverse of AB if
True False Problem
A = 1 5
Enter T or F depending on whether the statement is true or
and
false. (You must enter T or F  True and False will not work.)
BI/ = 1
1. If A is a square matrix such that AA equals the 0 matrix,
then A must equal the 0 matrix.
2.
If AB is defined, then BA is also defined.
(AB)
=
1.(1 pt) Suppose that:
Enter or depending on whether the statement true or
A
and B
2
1
false (You must enter `orF. True and False will not work.)
2
2
3
Giver the following descriptions, determine the following ele
.1
mentary matrices and their inverses
If Aisa square matrix then there exists matrix such
that AB equals the identity matrix
2.
IfA and Bare both square matrices such that AB equals
a. The elementary matrix E1 multiplies the first row of by
BA qquals the identity matrix, then Bisthe inverse ma
1/3.
trix of
E1
E1
3. 1 pt) Consider the following systems
(a)
b The elementary matrix E2 multiplies the second row of A
by4
2x5y
219y = 3
E2
(b)
c. The elementary matrix E3 switches the first and second
rows
A.
E3
E3
(i) Find the inverse of the (common) coefficient matrix of the
d. The elementary matrix E4 adds times the first row of A
two systems
to
the_second row
of A.
E4
EA
e. The elementary matrix Es multiplies the second row ofB
by 1/4.
(ii) Find the solutions to the two systems by using the inverse
i.e. by evaluating where represents the right hand side
Es
'E5'
(i.e. B
for system and
for system
f. The elementary matrix E6 multiplies the third row of B by
(b)).
4.
Solution 1 system (a):
Solution system (b)
E6
=
Er'
9
2
g The elementary matrix Ez switches the first and third rows
4. pt)IfA
ofB
o

E7
E
Then
h. The elementary matrix Es adds times the third row of B
9
to
the
second row
of B.

(1 pt) If A
Es
Ez

Then A
2. (l pt)
True False Problem
6. (1 pt) square matrix is called permutation matrir if it
Ux

contains the entry exactly once in each row and each col
umn, with all other entries being 0. All permutation mair rices are
Find th solution
invertible Find the inverse of the following permutation matrix
x=

A=

10. pt) Find the LU factorization of A =
12
15
and use it to solve the system
4
3
x
4
4
=
12
15
27
7. (1 pt) Determine which of the following formulas hold for
all invertible matrices/ and B.
A
,
A invertible
(AB) A¹B¹
x 
*2

(IA)(I+A)=1A²
x3
(I+A)(I+A¹)=21+A+A
11. (1 pt) Solve for X. Dc not use decimal numbers your
answer. If there are fractions, leave them unevaluated.
ABA!=B
l'is invertible
(A+B)(AB)=A²B²
AB=BA
x
12. (1 pt) Solve for X. Dc not use decimal numbers your
8.(l pt)Find the LUfactorization of A=
answer. there are fractions, leave them unevaluated.
*[==][E==].
9.
(1
pt) Find the LU factorization of A
X
13. (1 pt) Let / x3 matrix and suppose we know that
2ay+laz5ay=0
where a1 a and a are the columns of A. Write non trivial
solution to the system Ax =0
x
IsA singular or nonsingular? Check the correct answer below.
you would first solve
A. The matrix A nonsingular because it square
matrix.
B. The matrix A nonsingular because the homoge
neous systems Ax 0has anontrivial solution
. C. The matrix. 1it singular because itisa square matrix
D. The matrix A is singular because the homogeneous
systems Ax 0 has anontrivial solution.
16. pt) Determine the following equivalen representations
14. (1 pt) The 2x elementary matrix E can be obtained
of the following system fequations:
from the identity matrix using the row operation n 3r2.
Find EA if
10x+4y=8
3x+3y=27
EA
a. Find the augmented matrix of the system
15. pt)Consider the following GaussJordan reduction


4
1
1
32
8
7
o
8

32
1
9.
Finel system
o
1

Q
k
E1A
E2E1A
E3E21 EAR
c. Use the inverse satisfy the following matrix equation.
Find
y

E1
d. Find matrices that satisfy the following matrix equation
x
ty

E2
c. The graph below show the lines determined by the twe equa
tions in our system:
E4
"
X
Write product of elementary matri
Find the coordinates of
P (
Finc coordinates of yintercept of the red line
A =(0 
Find coordinates of xintercept of the green line
(.00
1. (1 pt) Given the matrix A =
find its determi
2 3)
nant.
3 5 0 0
A=
The determinant of A is
4
3
8 9 7
1 0 4
det(A) =
2. (1 pt) Given the matrix
012
8
3
7
6
7
4
(a) find its determinant;
4
Your answer is :
9
(b) does the matrix have an inverse?
then det (A) =
Your answer is (input Yes or No):
9. (1 pt) Find the determinant of the matrix
1
3
3. (1 pt) If A =
3
1
M
0 2 0 2
then det (A)
=
and
0 1 2 0
det (M)
=
4. (1 pt) Find the determinant of the matrix
3
10. (1 pt) Find the determinant of the matrix
3
3
B
5 0 2
1 0 3 0 0
4 2 2
M
0 2 0 0 3
det(B) =
0 0 0 3 1
5. (1 pt) Determine all minors and cofactors of
1
2
6 9 1
det(M) =
A
0 7 7
6 4 7
11. (1 pt) Given the matrix
Mu
Au
a 1 6
M12 =
A12
=
A=
a 2 7
,
M13 =
A13 =
2 7 a
M21 =
A21 =
M22 =
A22 =
find all values of a that make the JA = 0. Enter the values of a
M23 =
A23 =
as a commaseparated list:
M31 =
A31 =
12. (1 pt) Find k such that the matrix
M32 =
A32 =
4
3
M33 =
A33 =
M
12 15
6. (1 pt) A square matrix is called a permutation matrix if
3+k 11
each row and each column contains exactly one entry 1, with all
is singular.
other entries being 0. An example is
k
1 0 0
13. (1 pt) Find the determinant of the n x n matrix A with
p=
0 0 1
6's on the diagonal, l's above the diagonal, and O's below the
0 1 0
diagonal.
Find the determinant of this matrix.
det(P)=
det(A)
1. (1 pt) If A and B are 3x3 matrices, det(A) = 5,
6. (1 pt) If a x 4 4 matrix A with rows v1, 12, V3, and V4 has
det(B) = 4, then
determinant detA =
det (AB)
2v1+2v4
det 2A) =
then det
V2
det(A7) =
V3
det(B1) =
7v1+4v4
det (B2) =
7. (1 pt) Are the following statements true or false?
2. (1 pt) If the determinant of a matrix A is det(A) = 5.
and the matrix B is obtained from A by multiplying the first row
1. det(AT) = (1)det(A).
by 6, then det (B) = 
?
2. If two row interchanges are made in sucession, then the
determinant of the new matrix is equal to the determi
3. (1 pt) If the determinant of a 5 X 5 matrix A is det(A) = 2,
nant of the original matrix.
and the matrix C is obtained from A by swapping the first and
3.
If det(A) is zero, then two rows or two columns are the
third rows, then det (C) =
same, or a row or a column is zero.
4. (1 pt) If the determinant of a 5 X 5 matrix A is det(A) = 2,
4. The determinant of A is the product of the diagonal en
tries in A.
and the matrix D is obtained from A by adding 3 times the third
row to the first, then det (D) =
a
b

c
8. (1 pt) Given det
d
e
= 5, find the following
5. (1 pt) Suppose that a 4 X 4 matrix A with rows V1, v2, v3,
8
h
;
and V4 has determinant detA = 9. Find the following determi
determinants.
nants determinants:
g
h
i
9v1
det
a b c
=
det
V2
d e
13
a
b
c
V4
det
5d+a
Sebb
5f+
V2
8
h
i
V3
det
5d+a
5e+b
5f
V4
det
d
e
VI
8
h
;
VI 5v3
det
V2
1 0 1
V3
9. (1 pt) If B =
2 1 1
V4
2 1 2
then det (B³ =

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