1. Chap. 2 Matrices 2.1 Operations with Matrices
2.2 Properties of Matrix Operations
2.3 The Inverse of a Matrix
2.4 Elementary Matrices
2.5 Applications of Matrix Operations

2. 2.1 Operations with Matrices
Matrix representations:
An uppercase case: A, B, C, …
A representative element enclosed in brackets: [aij], [bij]
A rectangular array of numbers:
Vector (column/row matrix): boldface lowercase a1, a2, …, an
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Chapter 2

3. Section 2-1
Definitions
Equality of Matrices Two matrices A = [aij] and B = [bij] are equal if they have the same size (mn) and aij = bij for 1  i  m and 1  j  n.
Matrix Addition If A = [aij] and B = [bij] are matrices of size mn, then their sum is the mn matrix given by A+B = [aij + bij]. The sum of two matrices of different sizes is undefined.
Scalar Multiplication If A = [aij] is an mn matrix and c is a scalar, then the scalar multiplication of A by c is the mn matrix given by cA = [caij]
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4. Example 1 Consider the four matrices
Section 2-1
Example 1
Consider the four matrices
Matrices A and B are not equal because they are of different sizes.
Similarly, B and C are not equal.
Matrices A and D are equal if and only if (iff) x = 3
Remark: “p if and only if q” means that p implies q and q implies p.
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5. Subtraction of Matrices
Section 2-1
Subtraction of Matrices
If A and B are of the same size, AB represents the sum of A and (B). That is, AB = A+(1)B = [aij  bij].
cA  dB = [caij  dbij].
Example 3:
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6. Matrix Multiplication
Section 2-1
Matrix Multiplication
If A = [aij] is an mn matrix and B = [bij] is an np matrix, then the product AB is an mp matrix AB = [cij], where
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7. Example 4 Find the product AB, where and Section 2-1 Ming-Feng Yeh
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8. Section 2-1
Example 5
Matrix multiplication is not, in general, commutative.
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Chapter 2

9. Systems of Linear Equations
Section 2-1
Systems of Linear Equations
Matrix Equation: Ax = b
A: coefficient matrix; x and b: column matrix (vector)
Example 6: Solve the matrix equation Ax = 0, where
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10. Diagonal Matrix & Trace (p. 58)
Section 2-1
Diagonal Matrix & Trace (p. 58)
A square matrix is called a diagonal matrix if all entries that not on the main diagonal are zero.
The trace of an nn matrix A is the sum of the main diagonal entries. That is,
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11. 2.2 Properties of Matrix Operations
Theorem 2.1 Properties of Matrix Addition and Scalar Multiplication
If A, B, and C are mn matrices and c and d are scalars, then the following properties are true.
1. A+B = B+A Commutative property of addition
2. A+(B+C) = (A+B)+C Associative property of addition
3. (cd)A = c(dA) Associative property of multiplication
4. 1A = A Multiplication identity
5. c(A+B) = cA + cB Distributive property
6. (c+d)A = cA + dA Distributive property
交換律
結合律
結合律
乘法單位元素
左分配律
右分配律
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12. Section 2-2
Proof of Theorem 2.1
The proofs follow directly from the definitions of matrix addition and scalar multiplication, and the corresponding properties of real numbers.
Let A = [aij] and B = [bij]
1. Use the commutative properties of addition of real numbers to write A+B = [aij+bij] = [bij+aij] = B+A
5. Use the distributive properties (for real number) of multiplication over addition to write c(A+B) = [c(aij+bij)] = [caij+cbij] = cA+cB
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13. Zero Matrix & Additive Identity
Section 2-2
Zero Matrix & Additive Identity
If A is an mn matrix and Omn is the mn matrix consisting entirely of zeros, then A + Omn = A.
The matrix Omn is called a zero matrix, and it serves as the additive identity for the set of all mn matrices.
Theorem 2.2: Properties of Zero Matrix If A is an mn matrix and c is a scalar, then the following properties are true. 1. A + Omn = A. 2. A + (A) = Omn.  A is the additive inverse of A. 3. If cA = Omn, then c = 0 or A = Omn.
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14. Matrix Equation Real Numbers m  n Matrices
Section 2-2
Matrix Equation
Real Numbers m  n Matrices
Ex. 2: Solve for X in the equation 3X+A = B, where
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15. Theorem 2.3 Properties of Matrix Multiplication
Section 2-2
Theorem 2.3
Properties of Matrix Multiplication
If A, B, and C are matrices (with sizes such that the given matrix products are defined) and c is a scalar, then the following properties are true. 1. A(BC) = (AB)C Associative property 2. A(B+C) = AB + AC Distribution property 3. (A+B)C = AC + BC Distribution property 4. c(AB) = (cA)B = A(cB)
Proof of Property 2: A: mn matrix, B: np matrix, C: np matrix.
The entry in the ith row and jth column of A(B+C) is
The entry in the ith row and jth column of AB +AC is
equal
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16. Section 2-2
Noncommutativity
A commutative property for matrix multiplication was NOT listed in Theorem 2.3.
If A is of size 23 and B is of size 33, then the product AB is defined, but the product BA is not.
Example 4: Show that AB and BA are not equal for the matrices and
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17. Cancellation Property
Section 2-2
Cancellation Property
It does NOT have a general cancellation property for matrix multiplication.
If AC = BC, it is NOT necessary true that A = B.
Example 5: Show that AC = BC.
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18. Identity Matrix & Theorem 4
Section 2-2
Identity Matrix & Theorem 4
A square matrix that has 1’s on the main diagonal and 0’s elsewhere.
The identity matrix of order n:
Theorem 2.4: Properties of the Identity Matrix If A is a matrix of size mn, then the following properties are true. 1. AIn = A. 2. ImA = A.
If A is a square matrix of order n, then AIn = InA = A.
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19. Repeated Multiplication
Section 2-2
Repeated Multiplication
Repeated multiplication of a square matrix: For a positive integer k, Ak is
A0 = In, where A is a square matrix of order n.
Example 3: Find A3 for the matrix
k factors
j and k are nonnegative integer.
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20. Theorem 2.5 Number of Solutions of a System of Linear Equations
Section 2-2
Theorem 2.5
Number of Solutions of a System of Linear Equations
For a system of linear equations in n variables, precisely one of the following is true.
1. The system has exactly one solution.
2. The system has an infinite number of solutions.
3. The system has no solution.
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21. The Transpose of a Matrix
Section 2-2
The Transpose of a Matrix
The transpose of a matrix is formed by writing its columns as rows.
A matrix A is symmetric if A = AT.  aij = aji,  i  j.  a symmetric matrix must be square.
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22. Theorem 2.6 Properties of Transpose
Section 2-2
Theorem 2.6
Properties of Transpose
If A and B are matrices (with sizes such that the given matrix products are defined) and c is a scalar, then the following properties are true Transpose of a transpose Transpose of a sum Transpose of a scalar multiplication Transpose of a product
For any matrix A, the matrix is symmetric.
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23. Example 9 Show that are equal. Sol: Section 2-2 Ming-Feng Yeh
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24. Section 2-2
Example 10
For the matrix find the product and show that it is symmetric.
Sol:
Since , is symmetric.
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25. 2.3 The Inverse of a Matrix
Definition of an Inverse of a Matrix An nn matrix A is invertible (or nonsingular) if there exists an nn matrix B such that AB = BA = In
In is the identity matrix of order n.
The matrix B is called the (multiplicative) inverse of A.
A matrix that does NOT have an inverse is called noninvertible (or singular).
Nonsquare matrices do NOT have inverse.
Theorem 2.7: Uniqueness of an Inverse Matrix If A is an invertible matrix, then its inverse is unique. The inverse of A is denoted by
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26. Section 2-3
Proof of Theorem 2.7
Because A is invertible, it has at least one inverse B such that AB = BA = I.
Suppose that A has another inverse C such that AC = CA = I.
Then you can show that B and C are equal as follows. AB = I  C(AB) = C(I)  (CA)B = C  (I)B = C  B = C
Consequently B = C, and it follows that the inverse of a matrix is unique.
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27. Example 2 Find the inverse of the matrix
Section 2-3
Example 2
Find the inverse of the matrix
Sol: To find the inverse of A, try to solve the matrix equation AX = I for X.
Using matrix multiplication to check the result.
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28. Gauss-Jordan Elimination
Section 2-3
Gauss-Jordan Elimination
 The same coefficient matrix
(4)
[ A ┇ I ]  …  [ I ┇ A1 ]
Double augment matrix
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29. Procedure Let A be a square matrix of order n.
Section 2-3
Procedure
Let A be a square matrix of order n.
Write the n2n matrix [ A┇I ] (adjoining the matrices A & I)
If possible, row reduce A to I using elementary row operations on the entire matrix [ A┇I ]. The result will be the matrix [I ┇A1 ]. If this in not possible, then A is not invertible.
Check your work by multiplying to see that
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30. Example 4 Show that the matrix has no inverse. pf: (3) 2
Section 2-3
Example 4
Show that the matrix has no inverse.
pf:
(3) 2
It is not possible to rewrite [A┇I ] in the form [I ┇A1 ].
Hence A has no inverse.
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31. The Inverse of a 22 Matrix
Section 2-3
The Inverse of a 22 Matrix  +
The matrix A is a 22 matrix given by
The matrix A is invertible if and only if  = ad  bc  0.
If   0, then
pf:
Interchanging the entires on the main diagonal and changing the signs of the other two entires.
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32. Example 5 If possible, find the inverse of each matrix.
Section 2-3
Example 5
If possible, find the inverse of each matrix.
The matrix B is not invertible.
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33. Theorem 2.8 Properties of Inverse Matrix
Section 2-3
Theorem 2.8
Properties of Inverse Matrix
If A is an invertible matrix, k is a positive integer, and c is a scalar, then A1, Ak, cA, and AT are invertible and the flowing are true
Hint: if BC = CB = I, then C is the inverse of B.
pf: 1. Observe that , which means that A is the inverse of A1. Thus, 3.
Hence is the inverse of (cA), which implies that
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34. Section 2-3
Example 6
Compute A2 in two different ways and show that the results are equal.
1. (A2)1:
2. (A1)2:
the same result
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35. Theorem 2.9 The Inverse of a Product
Section 2-3
Theorem 2.9
The Inverse of a Product
If A and B are invertible matrices of size n, then AB is invertible and (AB)1 = B1A1.
pf: 1. (AB)(B1A1) = A(BB1)A1 = A(I)A1 = AA1 = I. 2. (B1A1)(AB) = B1(A1A)B = B1(I)B = B1B = I. Hence AB is invertible.
: in reverse order
Recall: (AB)T = BTAT
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36. Example 7 Find (AB)1 for the matrices and
Section 2-3
Example 7
Find (AB)1 for the matrices and
using the fact that A1 and B1 are given by
Sol:
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37. Theorem 2.10 Cancellation Property
Section 2-3
Theorem 2.10
Cancellation Property
If C is an invertible matrix, then the following properties hold.
1. If AC = BC, then A = B. Right cancellation property
2. If CA = CB, then A = B. Left cancellation property
pf: Use that fact that C is invertible and write AC = BC  (AC)C1 = (BC)C  A(CC1) = B(CC1)
 AI = BI
 A = B
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38. Theorem 2.11 Systems of Equations with Unique Solutions
Section 2-3
Theorem 2.11
Systems of Equations with Unique Solutions
If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by x = A1b.
pf: Ax = b  A1(Ax) = A1(b)  A1Ax = A1b  x = A1b
Example 8: Use an inverse matrix to solve each system
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39. 2.4 Elementary Matrices
Definition: An nn matrix is called an elementary matrix if it can be obtained from the identity matrix In by a single elementary row operation.
Elementary row operations: 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row.
The identity matrix In is elementary.  it can be obtained from itself by multiplying any one of its row by 1.
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40. Example 1 Which matrices are elementary?
Section 2-4
Example 1
Which matrices are elementary?
: (3)R2  R2
: it is not a square matrix
: (0)R3  R3 must be by a nonzero const.
: R2  R3
: R2+R1 R2
: two elementary row operations are required.
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41. Example 2 Elementary Matrices & Elementary Row Operations
Section 2-4
Example 2
Elementary Matrices & Elementary Row Operations
Interchange two rows: R1  R2
Multiply a row by a nonzero constant: (0.5)R2  R2
Add a multiple of a row to another row: R2+(2)R1  R2
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42. Theorem 2.12 Representing Elementary Row Operations
Section 2-4
Theorem 2.12
Representing Elementary Row Operations
Let E be the elementary matrix obtained by performing an elementary row operation on Im. If the same elementary row operation is performed on an mn matrix A, then the resulting matrix is given by the product EA.
Most applications of elementary row operations require a sequence of operations.  Gaussian elimination
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43. Section 2-4
Example 3
Find a sequence of elementary matrices that can be used to write the matrix A in row-echelon form.
Sol:
The elementary matrix E is a 33 matrix.
(2)
(1/2)
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44. Row Equivalence & Thms. 2.13~14
Section 2-4
Row Equivalence & Thms. 2.13~14
Definition: Let A and B be mn matrices. Matrix B is row-equivalent to A if there exists a finite number of elementary matrices E1, E2, …, Ek such that B = EkEk1E2E1A.
Theorem 2.13: Elementary Matrices Are Invertible If E is an elementary matrix, then E1 exists and is an elementary matrix.
Theorem 2.14: A Property of Elementary Matrices A square matrix A is invertible if and only if it can be written as the product of elementary matrices.
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45. Elementary Matrices Are Invertible
Section 2-4
Elementary Matrices Are Invertible
Elementary Matrix Inverse Matrix
R1  R2
R1  R2
R3+(2)R1 R3
R3+(2)R1 R3
(½)R3 R3
(2)R3 R3
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46. Section 2-4
Proof of Theorem 2.14
() If A can be written as the product of elementary matrices, then A is invertible.
pf: Assume that A is the product of elementary matrices. Then, because every elementary matrix is invertible and the product of invertible matrices is invertible, it follows that A is invertible.
() If A is invertible, then it can be written as the product of elementary matrices.
pf: Assume that A is invertible. The system of linear equations AX = O has only the trivial solution. This implies that [A┇O] can be rewritten in the form [I┇O] using the elementary operations as EkEk1E2E1A = I. Then it follows that Thus A can be written as the product of elementary matrices.
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47. Example 4 Find a sequence of elementary matrices whose product is Sol:
Section 2-4
Example 4
(1)
Find a sequence of elementary matrices whose product is
Sol:
(3)
(½)
(2)
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48.  Theorem 2.15 Equivalent Conditions
Section 2-4
 Theorem 2.15
Equivalent Conditions
If A is an nn matrix, then the following statements are equivalent.
1. A is invertible.
2. Ax = b has a unique solution for every n1 column vector b.
3. Ax = O has only the trivial solution.
4. A is row-equivalent to In.
5. A can be written as the product of elementary matrices.
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49. Section 2-4
LU-Factorization
A square matrix A is expressed as a product A = LU, where the square matrix L is lower triangular and the square matrix U is upper triangular.
Step 1: EkE2E1A = U : row reduction
Step 2:
Step 3: A = LU
By row reducing A
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50. Example 6 Find the LU-factorization of the matrix Sol: = U = L
Section 2-4
Example 6
(2)
Find the LU-factorization of the matrix
Sol: 4
= U
= L
 LU = A
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51. Solving a Linear System
Section 2-4
Solving a Linear System
Using LU-factorization to solve the linear system Ax = b: Ax = b and A = LU  LUx = b Let Ux = y.  Ly = b 1. Solve Ly = b for y. (forward substitution) 2. Solve Ux = y for x. (back-substitution)
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52. Example 7 Solve the linear system.
Section 2-4
Example 7
Solve the linear system.
Sol:
1. Let y = Ux and solve Ly = b for y
2. Solve Ux = y for x
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