Chapter 2

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## Chapter 2

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**Chapter 2**Matrices Gareth Williams J & B, 滄海書局**2.1 Addition, Scalar Multiplication, and Multiplication of**Matrices • 2.2 Properties of Matrix Operations • 2.3 Symmetric Matrices • 2.4 Inverse of a Matrix and Cryptography • 2.5 Leontief Input-Output Model • 2.6 Markov Chains, Population Movements, and Genetics • 2.7 A Communication Model and Graph Relationships in Sociology**2.1 Addition, Scalar Multiplication, and Multiplication of**Matrices Example 1 Determine the elements a12, a23, and a34 for the following matrix A. Given the main diagonal of A. Solution**Equality of Matrices**Definition The matrices are equal if they are of the same size and if their corresponding elements are equal. Thus A = B if aij = bij.**Addition of Matrices**Definition Let A and B be matrices of the same size. Their sumA + B is the matrix obtained by adding together the corresponding elements of A and B. The matrix A + B will be of the same size as A and B. If A and B are not of the same size, they cannot be added, and we say that the sum does not exist. Thus if C = A + B, then cij = aij+ bij.**Determine A + B, and A + C, if the sum exist.**Example 2 Solution**Scalar Multiplication of Matrices**Definition Let A be a matrix and c be a scalar. The scalar multiple of A by c, denoted cA, is the matrix obtained by multiplying every element of A by c. The matrix cA will be the same size as A.**Example 3**Solution**Negation and Subtraction**Definition We now define subtraction of matrices in such a way that makes it compatible with addition, scalar multiplication, and negative. Let A – B = A + (–1)B**Negation and Subtraction**Example Solution**Multiplication of Matrices**Definition Let the number of columns in a matrix A be the same as the number of rows in a matrix B. The product AB then exists. The element in row i and column j of AB is obtained by multiplying the corresponding elements of row i of A and column j of B and adding the products. If the number of columns in A does not equal the number of row B, we say that the product does not exist.**Let C = AB,**Example 6 Solution**A**m r Size of a Product Matrix If A is an m r matrix and B is an r n matrix, then AB will be an m n matrix. B = AB r n m n**Example 7**If A is a 5 6 matrix and B is an 6 7 matrix. What is the size of the product of AB? Solution**Definition**A zero matrix is a matrix in which all the elements are zeros. A diagonal matrix is a square matrix in which all the elements not on the main diagonal are zeros. An identity matrix is a diagonal matrix in which every diagonal element is 1.**Theorem 2.1**Let A be m n matrix and 0mn be the zero m n matrix. Let B be an n n square matrix. 0n and In be the zero and identity n n matrices. Then A + 0mn = 0mn + A = A B0n = 0nB = 0n BIn = InB = B Proof**Example 8**Solution**Matrix Notation and Systems of Equations**A system of m linear equations in n variables as follows Let We can write the system of equations in the matrix form AX = B**Matrix Multiplication in Terms of Columns**(a)Let A be an m×n matrix and B be an n×r matrix. Let the column of B be the matrices b1, b2, …, br, . Write B=[b1, b2, …, br]. Thus (b)Let A be an m×n matrix and B be an n×r matrix. Let the row of A be the matrices a1, a2, …, am, .**(1)**(2) Partitioning of Matrices**2.2 Properties of Matrix Operations**Theorem 2.2 Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed. Properties of Matrix Addition and scalar Multiplication 1. A + B = B + A Commutative property of addition 2. A + (B + C) = (A + B) + C Associative property of addition 3. A + 0 = 0 + A = A (where0 is the appropriate zero matrix) 4. c(A + B) = cA + cB Distributive property of addition 5. (a + b)C = aC + bC Distributive property of addition 6. (ab)C = a(bC)**Theorem 2.2**Proof**Theorem 2.2**Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed. Properties of Matrix Multiplication 1. A(BC) = (AB)C Associative property of multiplication 2. A(B + C) = AB + AC Distributive property of multiplication 3. (A + B)C = AC + BCDistributive property of multiplication 4. AIn = InA = A (whereIn is the appropriate identity matrix) 5. c(AB) = (cA)B = A(cB) Note: AB BA in general. Multiplication of matrices is not commutative.**Theorem 2.2**Proof**Example 1**Solution**Example 2**Solution**Example 3**By Example 2, compare the number of multiplications involved in the two ways (AB)C and A(BC) of computing the product ABC. Solution**Note**If A is an m r matrix and B is r n matrix, the number of scalar multiplications involved in computing the product AB is mrn. If A is an m r matrix and B is r n matrix, the number of additions involved in computing the product AB is m(r-1)n.**Caution**• In algebra we know that the following cancellation laws apply. • If ab = ac and a 0 then b = c. • If pq = 0 then p = 0 or q = 0. • However the corresponding results are not true for matrices. • AB = AC does not imply that B = C. • PQ = 0 does not imply that P = 0 or Q = 0.**Powers of Matrices**Theorem 2.3 If A is an n n square matrix and r and s are nonnegative integers, then 1. ArAs = Ar+s. 2. (Ar)s = Ars. 3. A0 = In (by definition) Proof**Example 4**Solution**Example 5**Simplify the following matrix expression. Solution**2.3 Symmetric Matrices**Definition The transpose of a matrix A, denoted At, is the matrix whose columns are the rows of the given matrix A.**Example 1**Solution**Theorem 2.4 Properties of Transpose**Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed. 1. (A + B)t = At + Bt Transpose of a sum 2. (cA)t = cAt Transpose of a scalar multiple 3. (AB)t = BtAt Transpose of a product 4. (At)t = A Proof**match**match Definition A symmetric matrix is a matrix that is equal to its transpose.**Example 3**Let A and B be symmetric matrices of the same size. Prove that the product AB is symmetric if and only if AB = BA. *We have to show (a) if AB is symmetric, then AB = BA, and the converse, (b) AB = BA, then AB is symmetric. Proof (a) Let AB be symmetric, then AB= by definition of symmetric matrix = by the transpose of a product = since A and B are symmetric matrices (b) Let AB = BA, then (AB)t = by the transpose of a product = since A and B are symmetric matrices =**Definition**Let A be a square matrix. The trace A, denoted tr(A) is the sum of the diagonal elements of A. Thus if A is an n n matrix. tr(A) = a11 + a22 + … + ann**Example 4**Determine the trace of the matrix Solution**Theorem 2.5 Properties of Trace**Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed. 1. tr(A + B) = tr(A) + tr(B) 2. tr(AB) = tr(BA) 3. tr(cA) = trc(A) 4. tr(At) = tr(A) Proof**Equality:**Addition: Subtraction: Multiplication: Matrices with Complex Elements The element of a matrix may be complex numbers. A complex number is of the form z = a + bi Where a and b are real numbers and a is called the real part and b the imaginary part of z. Let be the complex numbers. The rules of arithmetic for complex numbers are as follows.**Conjugate of a Complex Number**Definition The conjugate of a complex number z = a+bi is defined and written = a-bi**Consider the complex numbers**Compute Example 5 Solution