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8/3/2019 Matrices Determinants Slides
http://slidepdf.com/reader/full/matrices-determinants-slides 1/29
Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Theoretical Models in Computing
Dr. Mai Duc Thanh1
1Department of Mathematics, International University of Hochiminh City
Lecture 2: Matrices and Determinants
Dr. Mai Duc Thanh Theoretical Models in Computing
8/3/2019 Matrices Determinants Slides
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Outline
1 Matrices
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Outline
1 Matrices
2 Determinants
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Outline
1 Matrices
2 Determinants
3 The inverse matrix
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrices: Example
An animal breeder can buy three types of food for pigs. Eachcase of Brand A contains 25 units of fiber, 30 units of protein,and 30 units of fat. Each case of Brand B contains 50 units of fiber, 30 units of protein, and 20 units of fat. Each case of Brand C contains 75 units of fiber, 30 units of protein, and 20units of fat.How many cases of each should the breeder mix together to
obtain a food that provides 1,200 units of fiber, 600 units of protein, and 400 units of fat?
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrices: Example
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrices: Example
We have the following table
Brand A Brand B Brand C
Fiber 25 50 75Protein 30 35 40Fat 30 25 20
x cases y cases z cases
Let x , y , z represent number of cases of Brands A, B, C,
respectivelyTotal amount of fiber is to be 1,200 units:
25x + 50y + 75z = 1, 200. (1)
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrices: Example
Total amount of protein is 600:
30x + 35y + 40z = 600. (2)
Total amount of fat is 400:
30x + 25y + 20z = 400. (3)
Problem: find x , y , z satisfying the system
25x + 50y + 75z = 1, 20030x + 35y + 40z = 600
30x + 25y + 20z = 400.
(4)
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrices: Example
⇒ We need some data in the form
A =
25 50 75 1, 200
30 35 40 60030 25 20 400
.
Dr. Mai Duc Thanh Theoretical Models in Computing
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrices: Definition
Many practical applications of engineering and science,quantitative problems of business and economics, andmathematical models involve data of the form
A =
a11 a12 · · · a1n
a21 a22 · · · a2n
· · · · · · · · · · · ·am1 am2 · · · amn
(5)
This array of numbers enclosed by brackets is called an m × nmatrix with m rows and n columns. The entry aij denotes theelement in the ith row and jth column.
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Theoretical
Models inComputing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Square Matrices, Row and Column Vectors
If m = n then A is called a square matrix of order n
b =
b 1b 2...
b n
If A has only one column (n = 1) then A is called a columnvector
If A has only one row (m = 1) then A is called a row vector
c = (c 1, c 2, ..., c n)
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrices: Principle Diagonal
If A is a square matrix of order n, the diagonal containingelements a11, a22, ..., ann is called the principle, main orleading diagonal.
trace A = a11 + a22 + ...ann =n
i =1
aii
A diagonal matrix is a square matrix that has its onlynon-zero elements along the leading diagonal:
a11
0 0 · · · 00 a22 0 · · · 00 0 a33 · · · 0
· · · · · · · · · · · ·0 0 0 · · · ann
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Identity Matrix, Null Matrix
A special case of diagonal matrices: Unit matrix or identitymatrix I = I n for which a11 = a22 = ... = ann = 1:
I n =
1 0 0 · · · 00 1 0 · · · 00 0 1 · · · 0
· · · · · · · · · · · ·0 0 0 · · · 1
Zero or null matrix 0 is the matrix with every element zero
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Transpose Matrix
Transposed matrix AT of A by (5) is the matrix withelements aT
ij = a ji :
AT =
a11 a21 · · · an1
a12 a22 · · · an2· · · · · · · · · · · ·a1n a2n · · · amn
If a square matrix such that AT = A, then aij = a ji . So
elements are symmetric about main diagonal.If AT = A, then A is called a symmetric matrixIf AT = −A, then A is called a skew-symmetric orantisymmetric matrix
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Basic Operations of Matrices
(a) Equality: Two matrices A and B are equal if all their
elements are the same
A = B ⇔ aij = b ij , ∀1 ≤ i ≤ m, 1 ≤ j ≤ n.
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Basic Operations of Matrices
(b) Addition: Addition of matrices is straightforwardIf A = (aij ), B = (b ij ), 1 ≤ i ≤ m, 1 ≤ j ≤ n, then
a11 a12 · · · a1n
a21 a22 · · · a2n· · · · · · · · · · · ·
am1 am2 · · · amn
+
b 11 b 12 · · · b 1n
b 21 b 22 · · · b 2n· · · · · · · · · · · ·
b m1 b m2 · · · b mn
A + B =
a11 + b 11 a12 + b 12 · · · a1n + b 1n
a21 + b 21 a22 + b 22 · · · a2n + b 2n· · · · · · · · · · · ·
am1 + b m1 am2 + b m2 · · · amn + b mn
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Basic Operations of Matrices
(c) Multiplication by a scalar: Matrix λA has elements λaij :
A =
a11 a12 · · · a1n
a21 a22 · · · a2n
· · · · · · · · · · · ·am1 am2 · · · amn
then
λA =
λa11 λa12 · · · λa1n
λa21 λa22 · · · λa2n
· · · · · · · · · · · ·λam1 λam2 · · · λamn
(6)
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Basic Operations of Matrices
(d) Properties of transform:
(A + B )T
= AT
+ B T
, (AT
)T
= A
(e) Basic rules of addition:A + B = B + A
(A + B ) + C = A + (B + C )λ
(A + B ) =λ
A +λ
B
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Matrix Multiplication
If A is an m × p matrix with elements aij and B is a p × n
matrix with elements b ij , then we can define the product
C = AB as the m × n matrix with entries
c ij =
p k =1
aik b kj i = 1, 2, ..., m, j = 1, 2, ..., n
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Theoretical
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Example:
Given
A =
1 2 −10 3 2
, B =
−1 33 10 2
,
Find AB , BA.
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Properties of matrix multiplication
A(BC ) = (AB )C
(mA)B = A(mB ) = mAB (A + B )C = AC + BC , A(B + C ) = AB + AC
I mA = AI n = A
(AB )T = B T AT
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TheoreticalModels in
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Definition of Determinants
Given square matrices
A =
a11 a12
a21 a22
, B =
a11 a12 a13
a21 a22 a23
a31 a32 a33
Determinant of A, denoted by det A or |A|, is
|A| =
a11 a12
a21 a22
= a11a22 − a12a21
Determinant of 3 × 3 matrix B is
|B | = a11
a22 a23
a32 a33
− a12
a21 a23
a31 a33
+ a13
a21 a22
a31 a32
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TheoreticalModels in
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Definition of Determinants
Special case: if A = (a), then det A = a
Example: Evaluate the third-order determinant
1 2 −10 3 21 −1 0
Dr. Mai Duc Thanh Theoretical Models in Computing
C
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TheoreticalModels in
Computing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Determinants: Cofactors
If we take a determinant and delete row i and column j , thedeterminant remaining is called the minor M ij .In general, we take any row (or column) and evaluate an n × n
determinant |A| as
|A| =n
j =1
(−1)i + j aij M ij
Aij = (−1)i + j M ij is called the cofactor of element aij Thus,
|A| =
n j =1
aij Aij
Dr. Mai Duc Thanh Theoretical Models in Computing
P i f d i
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TheoreticalModels in
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Properties of determinants
(a) If two rows (or two columns) of a determinant are equal,then determinant is zero(b) Multiply a row (or a column) by a scalar:
|B | =λa11 λa12 λa13
a21 a22 a23
a31 a32 a33
= λ|A|
(c) Interchanging two rows (or two columns) changes the signof determinant
(d) Adding multiples of rows (or columns) together makes nodifference to the determinant(e) Transpose: |AT | = |A|(f) Product: |AB | = |A||B |
Dr. Mai Duc Thanh Theoretical Models in Computing
Adj i M i
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Matrices
Determinants
The inversematrix
Adjoint Matrices
The adjoint matrix of a matrix A is the transpose of the matrix
whose entries are cofactors of A:
adj A =
A11 A21 A31
A12 A22 A32
A13 A23 A33
Theorem: A(adjA) = |A|I .Thus,
|A||adj A| = |A(adj A)| = ||A|I n| = |A|n
If |A| = 0, this yields Cauchy theorem:
|adjA| = |A|n−
1
Alsoadj(AB ) = (adjB )(adj)A
If |A| = 0, then A is called singular. If |A| = 0, then A is called
nonsingular Dr. Mai Duc Thanh Theoretical Models in Computing
I f t i
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TheoreticalModels in
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Inverse of a matrix
Given a square matrix A, if we can construct a matrix B such
thatAB = BA = I
then we say: B is the inverse of A and write B = A−1
Since A(adjA) = |A|I , if |A| = 0 then
A−1 = 1|A|
adjA
If |A| = 0, then A−1 does not existIf the inverse A−1 exists, then it is unique.Indeed, if AC = CA = I , then
C = CI = C (AA−1) = (CA)A−1 = IA−1 = A−1
Inverse of a product:
(AB )−1 = B −1A−1
Dr. Mai Duc Thanh Theoretical Models in Computing
E l
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TheoreticalModels in
Computing
Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Example:
Let
1 2 2
2 1 22 2 1
show that A2 − 4A − 5I = 0 and hence A−1 = 15 (A − 4I ).
Calculate A−1 from this result.
Dr. Mai Duc Thanh Theoretical Models in Computing
H k A i t N2
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TheoreticalModels in
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Dr. Mai DucThanh
Matrices
Determinants
The inversematrix
Homework Assignment N2
Textbook: Glyn James, Modern Engineering Mathematics,Addition-Wesley, 2001.
-Pages 279-280: 5, 6, 8, 11, 14, 17-Pages 294: 20, 21, 22, 23, 26, 27, 30, 31-Pages 298-299: 34, 35, 36, 37Deadline: 31st March, 2009
Dr. Mai Duc Thanh Theoretical Models in Computing