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‫المملكة العربية السعودية‬
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‫جامعة المجمعة‬
‫كلية العلوم والدراسات األنسانية‬
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‫‪ 244‬ريض‬
‫‪Work Sheet :‬‬
‫‪Matrix Algebra‬‬
‫‪Math 244‬‬
‫‪Dr Alaa Kama‬‬
1
Question 1 : Let A , B and C be n×n square matrices.
Indicate true ( T ) or false ( F ) of the following.
1)
A B B A
(
)
2)
A B  B A
(
)
(
)
(
)
3)
4)
A B  BA
A (BC )  (A B )C
A (B  C )  A B +AC
1
6)
A 1 
A
7 ) A I  IA  A
5)
(
)
(
)
(
)
8)
If AC  AB , then B  C
9)
If AB  0, then A  0 or B  0 (
10 )
If A is invertible, then A 1A  AA 1  I (
11 )
If A and B are invertibles, then (AB )1  A 1B (
12 )
13 )
If A is invertible, then (A T ) 1  (A 1 )T (
tr(AB )  tr( A )  tr( B )
(
)
14 )
det(AB )  det(A )  det(B )
(
(
det(A  B )  det(A )  det(B )
1
16 ) det(A 1 )  (det(A )) 1 
det(A )
17 ) AB  A  B
15 )
)
)
)
)
)
)
(
)
(
)
(
)
18 )
If A is symmetric, then A 1 is symmetric
19 )
If AB  3I , then A is invertible
(
)
If A B  4I , then A 1  14 B
(
)
(
)
Dr Alaa Kama
2
20 )
Question 2:
Let
2
A
0

1
1
3 
2 
,
1 

4
0
2
2
1

1

2)
AB and BA
3)
AT , B T
4)
tr (A ), tr (B ) and tr (A  3B )
5)
and
and (AB )T
A 2 and B 2
6)
x
5
0
A  3B and B  2A
Calculat the following : 1)
Question 3: Find
1
B
 1

3
tr (A 2 ) and tr (B 2 )
y
so that
 2x  y
 7

3  10 3 

x  y   7 1
Question 4: Find a and b and c for which B is symmetric,
 6 a  3b a  c 
B   3
2
2 
 2 a  c
3 
 1 1 0 
 2 1 
Question 5: Let A   4 1 2  , B  



 3 0
 1 0 3
and P (x )  x 2  3x  2
Calculate P ( A ) and
P (B )
2 0
1
3
3
A

Question 6: Let
 4 1  , calculate A , A and A


Dr Alaa Kama
3
Question 7: Let
1
A   2
 1
0
3
1 
,B
4

1
2
3 
and
 3
C 
 4
0
1
1
5

1- Calculate AB , BC and AC
1 T
T
1
2- Calculate ABC , (B ) and (B )
Question 8: Let A  a 23 , B  b 32 and C  c 43
: Indicate true ( T ) or false ( F ) of the following.
1)
2)
AB  ab 33
(
)
(
)
BA  ba 33
3)
AC  ac 23
(
)
4)
CA  ca 43
(
)
5)
BC  bc 44
(
)
6)
CB  cb 42
(
)
1
A

Question 9: Let
0

10
0
and
2 
5
Calculate A , B , A
Question 10 :Let
1
A 
1

8
1 0 0 
B  0 2 0 
0 0 1
10
and B 5
0
2
0
5

3

4
By the cofactors Calculate
A 1 .
Question 11 : Solve the following system by Cramer's Rule :
2x
z  0 ,
x  3 y  2z  3 ,
Dr Alaa Kama
4
4x  2 y  3z  14
Question 12 : Solve the following system by the inverse of
A.
:
2 x  y  2 z  3 ,
x  3 y  2z  3 ,
3x  2 y  z  4
3 2 
A

Question 13 : Let
0 5


1- Calculate the characteristic polynomial of
2 - Calculate the Eigen-values of
A.
A.
1
3 - Using Cayley-Hamilton Theorem to give A .
Question 14 : Let
1
B 
1

8
0
2
4
0
5

3

1- Calculate the characteristic polynomial of
2 - Calculate the Eigen-values of
B.
B.
3 - Using Cayley-Hamilton Theorem to give B
1
.
Dr Alaa Kama