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Linear Algebra
Chapter 2
… part1
Matrices
S1
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
Definition
A matrix is a rectangular array of numbers.
The numbers in the array are called the elements of the matrix.
Denoted by: A,B,…
capital letter.
 a11 a12
a
a22
21

A


an 1 an 2
a1m 
a2 m 


anm 
Ch2_2
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
Note:
• aij: the element of matrix A in row….. and column ……
we say it is in the ………………..
• The size of a matrix = number of …... × number of ….….. = …….
• If n=m the matrix is said to be a …….. matrix with size = …... or ….
• A matrix that has one row is called a …… matrix.
• A matrix that has one column is called a ……….. matrix.
• For a square nn matrix A, the main diagonal is: ……………….
• We can denote the matrix by ……………..
Ch2_3
Example 1
2 5
A 

3
0


1) A size is ................
2) A is a ............... matrix .
3
B  1
0
it ' s
1 7 1
2 2 4
0 3 1 
size is .........
3) .............. is the main diagonal .
Definition
Two matrices are equal if:
1) …………………..
2) ……………………..
Ch2_4
Addition of Matrices
Definition
• If A and B be matrices of the …………….. then the
sum A + B=C will be of the ……….. size and
……………………
• IfA size  B size  A  B ..................
• Let A be a matrix and k be a scalar. The scalar multiple of A by
k , denoted ………… will be the same size as A.
……………………
•The matrix (-1)A= -A called the …………… of A.
•Let A and B of the same size then: A - B= A +(-B)=C and:
……………………
Ch2_5
Example 2
4 7, B   2 5  6, and C   5 4.
Let A   1
8
0  2 3
 3 1
 2 7
Determine A + B , 3A , A + C , A-B
Solution
(1) A  B 
(2) 3A 
(3) A  C 
(4) A  B 
Ch2_6
Definition
• A ……. matrix 0ij all of it’s elements are zero. If the zero
matrix is of a square size n×n it will be denoted by 0 n .
Theorem2.2:
Let A,B,C be matrices, k 1 , k 2 be scalars.
Assume that the size of the matrices are such that the operations
can be performed, let 0 be the zero matrix.
Properties of matrix addition and scalar multiplication:
1) A  B  ............
2) A   B  C   ...............
3) A  0  0  A  ........
4) k 1  A  B   .............
5)  k 1  k 2 C  .............
6) k 1  k 2 A   ................
Ch2_7
Example 3
Compute the linear combination: 2A  3B  5C
 1 3
 3 7 
0 2 
A 
,B  
,C  




4
5
2
1
3

1






for:
Solution
Ch2_8
Multiplication of Matrices
Definition
1) If the number of ……….. in A = the number of …….. in B.
The product AB then exists.
Let A: …….. matrix, B: ………. matrix,
The product matrix C=AB is a ………. matrix.
2) If the number of ………. in A  the number of …….. in B
then The product AB ……………..
Ch2_9
 a11 a12
a
a22
21

if A 


an 1 an 2
 a11 a12
a
a22
21

 AB 


an 1 an 2
a1m 
a2 m 


anm 
a1m 
a2 m 


anm 
 b11 b12
b
b 22
21

, B 


b m 1 b m 2
b1k
 b11 b12
b
b2k
 21 b 22


b mk
b m 1 b m 2
b1k 
b 2 k 


b mk 







Ch2_10
Note:
If
AB  C
then
c ij  ..............................................
Example 4
Let C = AB, A   2 1 and B   7 3 2 Determine c23.
 3 4
 5 0 1
Ch2_11
Example 5
 3 1 2
7 2 
Let A  
,B  


4
1
5
6
3




Solution
Determine AB , BA .
Note. In general, ……………
Ch2_12
Special Matrices
Definition
1) A …….. matrix is a matrix in which all the elements are zeros.
2) A ……….. matrix is a square matrix in which all the
elements ……………………………………...
3) An ……….. matrix is a diagonal matrix in which every
element in the main diagonal is …….
0mn
0
 0

0
0
0

0




0
0

0
a11 0
0 a
22
A 


0
0
0
0 


ann 
zero matrix
................. matrix A
1 0
0 1
In  


0 0
0
0 


1
............... matrix
Ch2_13
Theorem 2.1
Let A be m  n matrix and Omn be the zero m  n matrix. Let B be
an n  n square matrix. On and In be the zero and identity n  n
matrices. Then:
1) A + Omn = Omn + A = …….
2) BOn = OnB = ………
3) BIn = InB = ………
Example 6
Let A  2 1  3 and B   2 1.
8
4 5
 3 4
A O23  ...............
BO 2  .................
BI 2  ...........
Ch2_14
2.2 Algebraic Properties of Matrix Operations
Theorem 2.2 -2
Let A, B, and C be matrices and k be a scalar. Assume that the size of
the matrices are such that the operations can be performed.
Properties of Matrix Multiplication
1. A(BC) = ………….
Associative property of multiplication
2. A(B + C) = …………
Distributive property of multiplication
3. (A + B)C = …………
Distributive property of multiplication
4. AIn = InA =………
(where In is the identity matrix)
5. k(AB) = ………= ………
Note: AB BA in general.
Multiplication of matrices is not
commutative.
Ch2_15
Example 7
4

3, and C   1 .
Let A   1 2, B   0 1
  Compute ABC.
3  1
 1 0  2
 0
Solution
Ch2_16
Note:
In algebra we know that the following cancellation laws apply.
If ab = ac and a  0 then ………..
If pq = 0 then ……….. or ……….
However the corresponding results are not true for matrices.
AB = AC ………………. that B = C.
PQ = O ………………… that P = O or Q = O.
Example 8
 1 2
 1 2
 3 8 
(1) Consider the matrices A  
, B 
, and C  
.



2 4
 2 1
 3 2
Observe that .................................., but ................
 1 0
0 0 
(2) Consider the matrices P  
,
and
Q

.



0 0
0 1 
Observe that ........................, but ........................
Ch2_17
Powers of Matrices
Definition
If A is a square matrix and k is a positive integer, then
A  ...........................
k
Theorem 2.3
If A is an n  n square matrix and r and s are nonnegative
integers, then
1. ArAs = ……….
2. (Ar)s = ……….
3. A0 = ………
(by definition)
Ch2_18
Example 9
 1  2
4
If A  
,
compute
A
.

0
 1
Solution
Example 10 Simplify the following matrix expression.
A( A  2B)  3B(2 A  B)  A2  7 B 2  5 AB
Solution
Ch2_19
Idempotent and Nilpotent Matrices
Definition
A square matrix A is said to be:
• ………………. if …………..
• ………………. if there is a positive integer p s.t ……….…
The least integer p such that Ap=0 is called the
……………………. of the matrix.
Example 11
3 6 
(1) A  

1

2


3 9
(2) B  

1

3


Ch2_20
2.3 Symmetric Matrices
Definition
The …………….. of a matrix A, denoted ………, is the matrix
whose ………….. are the ………. of the given matrix A.
i.e., A : m  n  A t :..............
Example 12 Determine the transpose of the following matrices:
A   2 7, B   1 2  7, and C   1 3 4.
6
 8 0
4 5
Ch2_21
Theorem : Properties of Transpose
Let A and B be matrices and k be a scalar. Assume that the sizes
of the matrices are such that the operations can be performed.
1. (A + B)t = ………...
Transpose of a sum
2. (kA)t = ...…
Transpose of a scalar multiple
3. (AB)t = ………...
Transpose of a product
4. (At)t = ………...
Ch2_22
Symmetric Matrix
Definition
Let A be a square matrix:
1) If ………... then A called ………………... matrix.
2) If ………... then A called ………………... matrix.
Example 13
 2 5
 5 4 


symmetric matrices
match
 1
 0 ....... 4 
 0
,  1
7 ....... , 
 2
......
8
3

 4
0 2 4 
7
3 9 
3 2 3

9 3 6 
match
Ch2_23
Example 14
Let A and B be symmetric matrices of the same size.
C = aA+bB,
a,b are scalars. Prove that C is symmetric.
Proof
Ch2_24
Example 15
Let A and B be symmetric matrices of the same size.
Prove that the product AB is symmetric if and only if AB = BA.
Proof
Ch2_25
Example 16
Let A be a symmetric matrix. Prove that A2 is symmetric.
Proof
Ch2_26
Definition
Let A be a square matrix. The ………… of A denoted by …….. is
the …………………………………. of A.
Theorem : Properties of Trace .
Let A and B be matrices and k be a scalar. Assume that the sizes
of the matrices are such that the operations can be performed.
1. tr(A + B) = …………………..
2. tr(kA) = ………….
3. tr(AB) = …………
4. tr(At) = …………..
Ch2_27