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- 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 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 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.

Scalar Multiplication of Matrices 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 Matrices

Solution

Negation and Subtraction Matrices

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

Multiplication of Matrices 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.

Example 4 Matrices

Example 5 Matrices

A Matrices

m r

Size of a Product MatrixIf 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 Matrices

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 Matrices

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 Matrices

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

Theorem 2.1 Matrices

Example 8 Matrices

Solution

Matrix Notation and Systems of Equations Matrices

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 Matrices

(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, .

Example 9 Matrices

Example 9 Matrices

2.2 Properties of Matrix Operations Matrices

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 Matrices

Proof

Theorem 2.2 Matrices

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 Matrices

Proof

Example 1 Matrices

Solution

Example 2 Matrices

Solution

Example 3 Matrices

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 Matrices

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 Matrices

- 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 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 Matrices

Solution

2.3 Symmetric Matrices 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 Matrices

Solution

Theorem 2.4 Properties of Transpose Matrices

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

Example 3 Matrices

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 Matrices

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

Theorem 2.5 Properties of Trace Matrices

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

Theorem 2.5 Matrices

Equality: Matrices

Addition:

Subtraction:

Multiplication:

Matrices with Complex ElementsThe 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 Matrices

Definition

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

Conjugate Transpose of a Matrix Matrices

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*.

Example Matrices

Compute C*. Is C Hermitian?

Theorem 2.6 MatricesProperties of Conjugate Transpose

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

2.4 The Inverse of a Matrix Matrices

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.

Determine the Inverse of a Matrix Matrices

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]

Gauss-Jordan Elimination for Finding the Inverse of a Matrix Matrices

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.

Theorem Matrices

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

Properties of Matrix Inverse Matrices

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

Proof

Properties of Matrix Inverse Matrices

Example 4 Matrices

Solution

Let MatricesAX = 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.

Theorem 2.8Proof

Elementary Matrices Matrices

Definition

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

Elementary Matrices Matrices

Illustrated Example Matrices

Let

Interchange row 2 and row 3 of A:

Multiple row 2 by k:

Add k row 1 to row 2:

Theorem Matrices

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:

Theorem Matrices

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

Proof

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