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Chapter 2 … part1
Matrices
Linear Algebra
S 1
Ch2_2
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
DefinitionA 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.
11 12 1
21 22 2
1 2
m
m
n n nm
a a a
a a aA
a a a
Ch2_3
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
• 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 nn matrix A, the main diagonal is: ……………….• We can denote the matrix by ……………..
Note:
Ch2_4
Definition
Two matrices are equal if:
1) …………………..
2) ……………………..
Example 1
2 5
3 0
1) ................
2) ............... .
3) .............. .
A
A size is
A is a matrix
is the main diagonal
3 1 7 1
1 2 2 4
0 0 3 1
' .........
B
it s size is
Ch2_5
Addition of Matrices
Definition• If A and B be matrices of the …………….. then the
sum A + B=C will be of the ……….. size and
……………………
• If
• 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:
……………………
..................A size B size A B
Ch2_6
Example 2.72
45 and ,813652 ,320
741Let
CBA
Determine A + B , 3A , A + C , A-B
Solution
3
(3
(1)
(2)
)
(4)
A
A
A B
C
A B
Ch2_7
Definition• A ……. matrix all of it’s elements are zero. If the zero matrix is of a square size n×n it will be denoted by .
0ij
0n
Theorem2.2:Let A,B,C be matrices, 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 2,k k
1) ............
2) ...............
3) 0 0 ........
A B
A B C
A A
1
1 2
1 2
4) .............
5) .............
6) ................
k A B
k k C
k k A
Ch2_8
2 3 5A B C
Example 3
Compute the linear combination: for:
1 3 3 7 0 2, ,
4 5 2 1 3 1A B C
Solution
Ch2_9
Multiplication of MatricesDefinition1) 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_10
11 12 1 11 12 1
21 22 2 21 22 2
1 2 1 2
11 12 1 11 12 1
21 22 2 21 22 2
1 2 1 2
,
m k
m k
n n nm m m mk
m k
m k
n n nm m m mk
a a a b b b
a a a b b bif A B
a a a b b b
a a a b b b
a a a b b bAB
a a a b b b
Ch2_11
..............................................ij
If AB C
then c
Note:
Example 4
Let C = AB, Determine c23.
105
237 and 4312 BA
Ch2_12
Example 5
3 1 2 7 2Let , Determine , .
4 1 5 6 3A B AB BA
Solution
Note. In general, ……………
Ch2_13
Definition1) 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 …….
matrix zero
000
000000
mn0
11
22
0 0
0 0
0 0
................. matrix A
nn
a
aA
a
1 0 0
0 1 0
0 0 1
............... matrix
nI
Special Matrices
Ch2_14
Theorem 2.1Let 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
.4312 and 854
312Let
BA
23...............A O
2 .................BO
2...........BI
Ch2_15
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.
Theorem 2.2 -2
2.2 Algebraic Properties of Matrix Operations
Ch2_16
Example 7
.014
and ,201310 ,13
21Let
CBA Compute ABC.
Solution
Ch2_17
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.
Note:
Example 81 2 1 2 3 8
(1) Consider the matrices , , and .2 4 2 1 3 2
Observe that .................................., but ................
A B C
1 0 0 0(2) Consider the matrices , and .
0 0 0 1
Observe that ........................, but ........................
P Q
Ch2_18
Powers of Matrices
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)
Definition
If A is a square matrix and k is a positive integer, then
...........................kA
Ch2_19
Example 9
. compute ,01
21 If 4AA
Solution
Example 10 Simplify the following matrix expression.
ABBABABBAA 57)2(3)2( 22 Solution
Ch2_20
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.
3 6(1)
1 2A
Example 113 9
(2) 1 3
B
Ch2_21
2.3 Symmetric Matrices
Definition
The …………….. of a matrix A, denoted ………, is the matrix whose ………….. are the ………. of the given matrix A.
Example 12
.431 and ,654721 ,08
72
CBA
i.e., : : ..............tA m n A
Determine the transpose of the following matrices:
Ch2_22
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_23
Symmetric Matrix
1 0 2 40 ....... 4
2 5 0 7 3 9 , 1 7 ....... ,
5 4 2 3 2 3...... 8 3
4 9 3 6
match
match
DefinitionLet A be a square matrix:
1) If ………... then A called ………………... matrix.
2) If ………... then A called ………………... matrix.
Example 13 symmetric matrices
Ch2_24
Example 14
Proof
Let A and B be symmetric matrices of the same size.
C = aA+bB, a,b are scalars. Prove that C is symmetric.
Ch2_25
Example 15
Proof
Let A and B be symmetric matrices of the same size.
Prove that the product AB is symmetric if and only if AB = BA.
Ch2_26
Example 16
Proof
Let A be a symmetric matrix. Prove that A2 is symmetric.
Ch2_27
DefinitionLet A be a square matrix. The ………… of A denoted by …….. is the …………………………………. of A.
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) = …………..
Theorem : Properties of Trace .