Matrices

BRANCH-E&TC ENGINEERING

SEM – 3^rd

SUBJECT- Engg math-iii

CHAPTER– 2 – matrices

TOPIC- matrices

Ay-2021-2022 , winter-2021

FACULTY- Tapas si (faculty Mathematics)

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Matrices - Introduction

Matrix algebra has at least two advantages:

• Reduces complicated systems of equations to simple expressions
• Adaptable to systematic method of mathematical treatment and well suited to computers

Definition:

A matrix is a set or group of numbers arranged in a square or rectangular array enclosed by two brackets

Matrices - Introduction

Properties:

• A specified number of rows and a specified number of columns
• Two numbers (rows x columns) describe the dimensions or size of the matrix.

Examples:

3x3 matrix

2x4 matrix

1x2 matrix

Matrices - Introduction

A matrix is denoted by a bold capital letter and the elements within the matrix are denoted by lower case letters

e.g. matrix [A] with elements a_ij

i goes from 1 to m

j goes from 1 to n

A_mxn=

_mAⁿ

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Matrices - Introduction

TYPES OF MATRICES

1. Column matrix or vector:

The number of rows may be any integer but the number of columns is always 1

Matrices - Introduction

TYPES OF MATRICES

2. Row matrix or vector

Any number of columns but only one row

Matrices - Introduction

TYPES OF MATRICES

3. Rectangular matrix

Contains more than one element and number of rows is not equal to the number of columns

Matrices - Introduction

TYPES OF MATRICES

4. Square matrix

The number of rows is equal to the number of columns

(a square matrix A has an order of m)

m x m

The principal or main diagonal of a square matrix is composed of all elements a_ij for which i=j

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Matrices - Introduction

TYPES OF MATRICES

5. Diagonal matrix

A square matrix where all the elements are zero except those on the main diagonal

i.e. a_ij =0 for all i = j

a_ij = 0 for some or all i = j

Matrices - Introduction

TYPES OF MATRICES

6. Unit or Identity matrix - I

A diagonal matrix with ones on the main diagonal

i.e. a_ij =0 for all i = j

a_ij = 1 for some or all i = j

Matrices - Introduction

TYPES OF MATRICES

7. Null (zero) matrix - 0

All elements in the matrix are zero

For all i,j

Matrices - Introduction

TYPES OF MATRICES

8. Triangular matrix

A square matrix whose elements above or below the main diagonal are all zero

Matrices - Introduction

TYPES OF MATRICES

8a. Upper triangular matrix

A square matrix whose elements below the main diagonal are all zero

i.e. a_ij = 0 for all i > j

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Matrices - Introduction

TYPES OF MATRICES

A square matrix whose elements above the main diagonal are all zero

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8b. Lower triangular matrix

i.e. a_ij = 0 for all i < j

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Matrices – Introduction

TYPES OF MATRICES

9. Scalar matrix

A diagonal matrix whose main diagonal elements are equal to the same scalar

A scalar is defined as a single number or constant

i.e. a_ij = 0 for all i = j

a_ij = a for all i = j

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Matrices

Matrix Operations

Matrices - Operations

EQUALITY OF MATRICES

Two matrices are said to be equal only when all corresponding elements are equal

Therefore their size or dimensions are equal as well

A =

B =

A = B

Matrices - Operations

Some properties of equality:

• IIf A = B, then B = A for all A and B
• IIf A = B, and B = C, then A = C for all A, B and C

A =

B =

If A = B then

Matrices - Operations

ADDITION AND SUBTRACTION OF MATRICES

The sum or difference of two matrices, A and B of the same size yields a matrix C of the same size

Matrices of different sizes cannot be added or subtracted

Matrices - Operations

Commutative Law:

A + B = B + A

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Associative Law:

A + (B + C) = (A + B) + C = A + B + C

A

2x3

B

2x3

C

2x3

Matrices - Operations

A + 0 = 0 + A = A

A + (-A) = 0 (where –A is the matrix composed of –a_ij as elements)

Matrices - Operations

SCALAR MULTIPLICATION OF MATRICES

Matrices can be multiplied by a scalar (constant or single element)

Let k be a scalar quantity; then

kA = Ak

Ex. If k=4 and

Matrices - Operations

Properties:

• k (A + B) = kA + kB
• (k + g)A = kA + gA
• k(AB) = (kA)B = A(k)B
• k(gA) = (kg)A

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Matrices - Operations

MULTIPLICATION OF MATRICES

The product of two matrices is another matrix

Two matrices A and B must be conformable for multiplication to be possible

i.e. the number of columns of A must equal the number of rows of B

Example.

A x B = C

(1x3) (3x1) (1x1)

Matrices - Operations

B x A = Not possible!

(2x1) (4x2)

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A x B = Not possible!

(6x2) (6x3)

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Example

A x B = C

(2x3) (3x2) (2x2)

Matrices - Operations

Successive multiplication of row i of A with column j of B – row by column multiplication

Matrices - Operations

Remember also:

IA = A

Matrices - Operations

Assuming that matrices A, B and C are conformable for the operations indicated, the following are true:

1. AI = IA = A
2. A(BC) = (AB)C = ABC - (associative law)
3. A(B+C) = AB + AC - (first distributive law)
4. (A+B)C = AC + BC - (second distributive law)

Caution!

1. AB not generally equal to BA, BA may not be conformable
2. If AB = 0, neither A nor B necessarily = 0
3. If AB = AC, B not necessarily = C

Matrices - Operations

AB not generally equal to BA, BA may not be conformable

Matrices - Operations

If AB = 0, neither A nor B necessarily = 0

Matrices - Operations

TRANSPOSE OF A MATRIX

If :

2x3

Then transpose of A, denoted A^T is:

For all i and j

Matrices - Operations

To transpose:

Interchange rows and columns

The dimensions of A^T are the reverse of the dimensions of A

2 x 3

3 x 2

Matrices - Operations

Properties of transposed matrices:

1. (A+B)^T = A^T + B^T
2. (AB)^T = B^T A^T
3. (kA)^T = kA^T
4. (A^T)^T = A

Matrices - Operations

1. (A+B)^T = A^T + B^T

Matrices - Operations

(AB)^T = B^T A^T

Matrices - Operations

SYMMETRIC MATRICES

A Square matrix is symmetric if it is equal to its transpose:

A = A^T

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Matrices - Operations

When the original matrix is square, transposition does not affect the elements of the main diagonal

The identity matrix, I, a diagonal matrix D, and a scalar matrix, K, are equal to their transpose since the diagonal is unaffected.

Matrices - Operations

INVERSE OF A MATRIX

Consider a scalar k. The inverse is the reciprocal or division of 1 by the scalar.

Example:

k=7 the inverse of k or k^-1 = 1/k = 1/7

Division of matrices is not defined since there may be AB = AC while B = C

Instead matrix inversion is used.

The inverse of a square matrix, A, if it exists, is the unique matrix A^-1 where:

AA^-1 = A^-1 A = I

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Matrices - Operations

Example:

Because:

Matrices - Operations

Properties of the inverse:

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A square matrix that has an inverse is called a nonsingular matrix

A matrix that does not have an inverse is called a singular matrix

Square matrices have inverses except when the determinant is zero

When the determinant of a matrix is zero the matrix is singular

Matrices - Operations

DETERMINANT OF A MATRIX

To compute the inverse of a matrix, the determinant is required

Each square matrix A has a unit scalar value called the determinant of A, denoted by det A or |A|

If

then

Matrices - Operations

If A = [A] is a single element (1x1), then the determinant is defined as the value of the element

Then |A| =det A = a₁₁

If A is (n x n), its determinant may be defined in terms of order (n-1) or less.

Matrices - Operations

MINORS

If A is an n x n matrix and one row and one column are deleted, the resulting matrix is an (n-1) x (n-1) submatrix of A.

The determinant of such a submatrix is called a minor of A and is designated by m_ij , where i and j correspond to the deleted

row and column, respectively.

m_ij is the minor of the element a_ij in A.

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Matrices - Operations

Each element in A has a minor

Delete first row and column from A .

The determinant of the remaining 2 x 2 submatrix is the minor of a₁₁

eg.

Matrices - Operations

Therefore the minor of a₁₂ is:

And the minor for a₁₃ is:

Matrices - Operations

COFACTORS

The cofactor C_ij of an element a_ij is defined as:

When the sum of a row number i and column j is even, c_ij = m_ij and when i+j is odd, c_ij =-m_ij

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Matrices - Operations

DETERMINANTS CONTINUED

The determinant of an n x n matrix A can now be defined as

The determinant of A is therefore the sum of the products of the elements of the first row of A and their corresponding cofactors.

(It is possible to define |A| in terms of any other row or column but for simplicity, the first row only is used)

Matrices - Operations

Therefore the 2 x 2 matrix :

Has cofactors :

And:

And the determinant of A is:

Matrices - Operations

Example 1:

Matrices - Operations

For a 3 x 3 matrix:

The cofactors of the first row are:

Matrices - Operations

The determinant of a matrix A is:

Which by substituting for the cofactors in this case is:

Matrices - Operations

Example 2:

Matrices - Operations

ADJOINT MATRICES

A cofactor matrix C of a matrix A is the square matrix of the same order as A in which each element a_ij is replaced by its cofactor c_ij .

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Example:

If

The cofactor C of A is

Matrices - Operations

The adjoint matrix of A, denoted by adj A, is the transpose of its cofactor matrix

It can be shown that:

A(adj A) = (adjA) A = |A| I

Example:

Matrices - Operations

Matrices - Operations

USING THE ADJOINT MATRIX IN MATRIX INVERSION

Since

AA^-1 = A^-1 A = I

and

A(adj A) = (adjA) A = |A| I

then

Matrices - Operations

Example

A =

To check

AA^-1 = A^-1 A = I

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Matrices - Operations

Example 2

|A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2

The determinant of A is

The elements of the cofactor matrix are

Matrices - Operations

The cofactor matrix is therefore

so

and

Matrices - Operations

The result can be checked using

AA^-1 = A^-1 A = I

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The determinant of a matrix must not be zero for the inverse to exist as there will not be a solution

Nonsingular matrices have non-zero determinants

Singular matrices have zero determinants

Matrix Inversion

Simple 2 x 2 case

Simple 2 x 2 case

Let

and

Since it is known that

A A^-1 = I

then

Simple 2 x 2 case

Multiplying gives

It can simply be shown that

Simple 2 x 2 case

thus

Simple 2 x 2 case

Simple 2 x 2 case

Simple 2 x 2 case

Simple 2 x 2 case

So that for a 2 x 2 matrix the inverse can be constructed in a simple fashion as

• Exchange elements of main diagonal
• Change sign in elements off main diagonal
• Divide resulting matrix by the determinant

Simple 2 x 2 case

Example

Check inverse

A^-1 A=I

Matrices and Linear Equations

Linear Equations

Linear Equations

Linear equations are common and important for survey problems

Matrices can be used to express these linear equations and aid in the computation of unknown values

Example

n equations in n unknowns, the a_ij are numerical coefficients, the b_i are constants and the x_j are unknowns

Linear Equations

The equations may be expressed in the form

AX = B

where

and

n x n

n x 1

n x 1

Number of unknowns = number of equations = n

Linear Equations

If the determinant is nonzero, the equation can be solved to produce n numerical values for x that satisfy all the simultaneous equations

To solve, premultiply both sides of the equation by A^-1 which exists because |A| = 0

A^-1 AX = A^-1 B

Now since

A^-1 A = I

We get

X = A^-1 B

So if the inverse of the coefficient matrix is found, the unknowns, X would be determined

Linear Equations

Example

The equations can be expressed as

Linear Equations

When A^-1 is computed the equation becomes

Therefore

Linear Equations

The values for the unknowns should be checked by substitution back into the initial equations