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

2.1 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 Ming-Feng Yeh Chapter 2

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] Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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: Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

Example 4 Find the product AB, where and Section 2-1 Ming-Feng Yeh Chapter 2

Section 2-1 Example 5 Matrix multiplication is not, in general, commutative. Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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, Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. 1. Transpose of a transpose 2. Transpose of a sum 3. Transpose of a scalar multiplication 4. Transpose of a product For any matrix A, the matrix is symmetric. Ming-Feng Yeh Chapter 2

Example 9 Show that are equal. Sol: Section 2-2 Ming-Feng Yeh Chapter 2

Section 2-2 Example 10 For the matrix find the product and show that it is symmetric. Sol: Since , is symmetric. Ming-Feng Yeh Chapter 2

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 . Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

Gauss-Jordan Elimination Section 2-3 Gauss-Jordan Elimination  The same coefficient matrix (4) [ A ┇ I ]  …  [ I ┇ A1 ] Double augment matrix Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. 1. 2. 3. 4. 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 Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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: Ming-Feng Yeh Chapter 2

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1  A(CC1) = B(CC1)  AI = BI  A = B Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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) Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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. Ming-Feng Yeh Chapter 2

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) Ming-Feng Yeh Chapter 2

 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. Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2

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) Ming-Feng Yeh Chapter 2

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 Ming-Feng Yeh Chapter 2