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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

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.

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.

Solution

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.

Solution

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

Example

Solution

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,

Solution

A

m r

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

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.

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

Solution

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

(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)

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 + ACommutative property of addition

2. A + (B + C) = (A + B) + CAssociative property of addition

3. A + 0 = 0 + A = A(where0 is the appropriate zero matrix)

4. c(A + B) = cA + cBDistributive property of addition

5. (a + b)C = aC + bCDistributive property of addition

6. (ab)C = a(bC)

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)CAssociative 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.

Proof

Solution

Solution

By Example 2, compare the number of multiplications involved in the two ways (AB)C and A(BC) of computing the product ABC.

Solution

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.

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

Solution

Simplify the following matrix expression.

Solution

Definition

The transpose of a matrix A, denoted At, is the matrix whose columns are the rows of the given matrix A.

Solution

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 + BtTranspose of a sum

2. (cA)t = cAtTranspose of a scalar multiple

3. (AB)t = BtAtTranspose of a product

4. (At)t = A

Proof

match

match

Definition

A symmetric matrix is a matrix that is equal to its transpose.

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

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:

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.

Definition

The conjugate of a complex number z = a+bi is defined and written = a-bi

Consider the complex numbers

Compute

Solution

Compute A + B, 2A, and AB.

Solution

Definition

(1) The conjugate of a matrix A is denoted and is obtained by taking the conjugate of each element of the matrix.

(2) The conjugate transpose of A is written and denoted by

(3) A square matrix C is said be Hermitian if C = C*.

Compute C*. Is C Hermitian?

Let A and B be matrices with complex elements and let z be a complex number.

1. (A + B)* = A* + B* Conjugate transpose of a sum

2. (zA)* = zA* Conjugate transpose of a scalar multiple

3. (AB)* = B*A* Conjugate transpose of a product

4. (A*)* = A

Definition

Let A be an n n matrix. If a matrix B can be found such that AB = BA = In, then A is said to be invertible and B is called inverse of A. If such a matrix B does not exist, then A has no inverse.

Prove that the matrix has the inverse

Proof

The inverse of an invertible matrix is unique.

Proof

AA-1=A-1A=In

A-k=A-1A-1…A-1

k times

AA-1=In

A-1=X=[x1, x2, …,xn] and In=[e1, e2, …,en]

Where

AX=A[x1, x2, …,xn]= [Ax1, Ax2, …,Axn] =[e1, e2, …,en]

Thus Ax1=e1, Ax2=e2, … ,Axn=en.

[A : In] →……. →[In : A-1]

Let A be an n n matrix.

1. Adjoin the identity n n matrix In to A to form the matrix [A : In].

2. Compute the reduced echelon form of [A : In].

If the reduced echelon form is of the type [In : B], then B is the inverse of A.

If the reduced echelon form is not of the type [In : B], in that the first n n submatrix is not In, then A has no inverse.

An n n matrix A is invertible if and only if it is row equivalent to In.

Determine the inverse of the matrix

Solution

Determine the inverse of the following matrix, if it exist.

Solution

Let A and B be inverse matrices and c a nonzero scalar, Then

Proof

Solution

Let AX = B be a system of n linear equation in n variable. If A–1 exists, the solution is unique and is given by X = A–1B.

Proof

Solve the system of equation

Solution

Definition

An elementary matrix is one that can be obtained from the identity matrix In through a single elementary row operation.

Let

Interchange row 2 and row 3 of A:

Multiple row 2 by k:

Add k row 1 to row 2:

Each elementary matrix is square and invertible.

Interchange row 2 and row 3 of A:

Multiple row 2 by a:

Add c row 1 to row 2:

If A and B are row equivalent matrices and A is invertible, then B is invertible.

Proof