5.MATRIX ALGEBRA.ppt

8/10/2019 5.MATRIX ALGEBRA.ppt

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MATRIX

A matrix is rectangular array of elements.

Generally, any rectangular array of numbers

surrounded by a pair of brackets is called a matrix.

Each matrix has rows and columns and this defines thesize of the matrix.

If a matrix [A] has m rowsn

columns, the size of thematrix is denoted by mx n.

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The matrix [A] may also be denoted by

[A]m x n to show that [A] is a matrix with

mrow and ncolumn.

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SPECIAL TYPES OF MATRIX

1. VECTOR

A vector is a matrix that has only one row or onecolumn.

There are two types of vectors.

Row vector OR Row matrix

Column vector OR Column matrix

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ROW VECTOR/ROW MATRIXA matrix that consists of just one row and any

number of columns is called row matrix OR rowVector

COLUMN VECTOR/COLUMN MATRIXA matrix that consists of single column and any

number of rows is called a column matrix OR

column vector

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SUB MATRIX

If some row(s) or/ and column(s) of a

matrix are deleted, the remaining matrix is

called sub matrix of that particular matrix.

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SQUARE MATRIX

If the number of rows of a matrix is equal

to the number of columns of a matrix

(r = c) is called a square matrix.

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A square matrix has two diagonals.

Upper diagonal extending from theupper left hand corner to the lowerright hand corner

It is called principal diagonal or

main diagonal and its elements arecalled diagonal elements.

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UPPER TRIANGULAR MATRIX

A matrix, in which all the elements below the diagonalentries are zero is called upper triangular matrix.

LOWER TRIANGULAR MATRIX

A matrix, in which all the elements above the diagonal

entries are zero is called lower triangular matrix

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DIAGONAL MATRIX

A square matrix with all non diagonal elementsequal to zero is called a diagonal matrix.

That is, only the diagonal entries of the squarematrix can be non zero.

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UNIT MATRIX OR IDENTY MATRIX

A square matrix with all diagonal elements equalto one is called an identity matrix or unit matrix.

Here non-diagonal elements are equal to zero

NUL MATRIX OR ZERO MATRIX

A matrix (square or rectangular), every elements of

which is zero, is called a null matrix or zero matrix.

It is denoted by the symbol o. [o m x n]

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EQUALITY OF TWO MATRICES

Two matrices are said to be equal if and only if

They are of the same order

Each element of the first matrix is equal to the

corresponding element of the second matrix.

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TRANSPOSE OF A MATRIX


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TRANSPOSE OF A MATRIX

Let A = [aij] be a matrix of order m x n, then

the matrix of order n x m obtained byinterchanging the rows and columns ofmatrix.

A is called the transpose of A and is denotedby AI OR AT. The number of rows of A isthen the same as the number of columns of

ATand vice versa.

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PROPERTIES OF TRANSPOSE OF A MATRIX

i. The transpose of the transpose of a matrix is the matrix itself. Then. AAT T ii.

If A be any m xn matrix, then TkAA Tk , where k is a non-zero scalar.iii. If A and B are two matrices of order m xn, then TT BABA T , the

transpose of the sum of two matrices is equal to the sum of their transpose.

ADDITION /SUBSTRACTION OF MATRICES


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ADDITION /SUBSTRACTION OF MATRICES

Let A = [aij] and B = [bij] be two matrices of the

same order m x n

their sum(differences) to be denoted by

A + B (A + B), is defined to be the matrixC = [cij] of order m x n,

where each element of Cis the sum (difference)of the corresponding elements of A and B, taken inthat order [cij= aij + bij] OR [cij= aij- bij].

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PROPERTIES OF MATRIX ADDITION


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PROPERTIES OF MATRIX ADDITION

i. Matrix addition is commutative. If A and Bbe two m x n matrices, then A+B=B+ A.

ii. Matrix addition is associative. If A, B, C, be

three matrices conformable for addition,then (A+B) +C = A+ (B+C).

iii. Existence of additive identity. If Abe m x nmatrix and O be also m x n zero matrix,then A+O = A = O+A.

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MULTIPLICATION OF A MATRIX BY A SCALAR ORSCALAR MULTIPLICATION

Let Abe any m x n matrix and kbe any real

OR complex number called scalar.

Then m x n matrix obtained by multiplying

every element of the matrix Aby a scalar kiscalled the scalar multiple of A by k and isdenoted by kA or Ak.

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MATRIX MULTIPLICATION

To multiply matrices, it is not necessary that they be of thesame order.

The requirement is that the number of columns of the f irstmatr ix be the same as the number of rows of the secondmatrix.

Matrices that satisfy this requirement are said to beconformablefor matrix multiplication.

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D t i th f ki l t i M d


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Determine the revenue of a parking lot on a given Monday,Tuesday, and Wednesday based on the following data.

The rupees charge per vehicle is Rs.4/= for Cars and Rs.8/=for buses.

Calculate the revenue per day.

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DAYS No.of.Car No.of.BusMonday 30 5

Tuesday 25 5

Wednesday 35 15


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DETERMINANTS OF THE MATRIX

The determinant is a single number or scalar and is found only for square matrices.

i. SINGULAR

If the determinant of a matrix is equal to zero, the matrix is termed singular. That is,

A = 0.

ii. NON SINGULAR

If the determinant of a matrix is not equal to zero, the matrix is termed non-singular.

That is, A 0


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SECOND ORDER DETERMINANT

The determinant Aof a 2 x 2 matrix called second order determinant. It is derived by taking

the product of the two elements on the principal diagonal and subtracting from it the productelements off the principal diagonal.

A= 211222112221

1211 aaaa

aa

aa


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THIRD ORDER DETERMINAT

The determinant of a 3 x 3 matrix can be calculated as follows,

333231

232221

131211

aaa

aaa

aaa

A


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Each element in a square matrix has its own minor. The minoris the value of the determinant of the matrix that results fromcrossing out the row and column of the element under

consideration.

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MINORS & COFACTORS

i. MINOR

A minor of the given matrix is the determinant of any of its square sub-matrix. Thus, a

minor ijM is the determinant of the sub matrix formed by deleting the i th row and

j th column of the matrix.


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i COFACTOR


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Cofactors

Each element in a square matr ix has its own cofactor. The

cofactor is the product of the elementsplace sign and

minor.26

i.

COFACTOR

A cofactor ijC is a minor with a prescribed sign. The rule for the sign of a cofactor

is

ijC = IJji M )1(

If the sum of subscripts (i + j ) is an even number, ijC = ijM . Since -1,

raise to an even power is positive.

If i + j is equal to an odd number ijC = - ijM . Since, -1 raised to an odd

power is negative.

INVERSE OF MATRIX


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INVERSE OF MATRIX

Inverse of a matrix can be found only for a square matrix.

The inverse of a matrix [A] is denoted by [A]-1.

The product of a matrix and its inverse results in an identity

matrix [I].

The identity matrix [I] has one for the diagonal elements

and all off-diagonal elements are zero

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MATRIX EXPRESSION OF SYSTEM OF LINEAR EQUATION

Matrix algebra permits the concise expression of a system of linear equations. Consider the

following example.

This can be expressed in matrix form2954

4537

21

21

xx

xx


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AX = B

54

37A

2

1X

x

xand

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45B

Here

i.

A is the coefficient matrix

ii.

X is the solution vector

iii.

B is the vector of constant termsiv.

A and B will always be column vectors


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MATRICESSOLVING TWO SIMULTANEOUS EQUATIONS

One of the most important applications of matrices is to the solution of linear simultaneousequations. Consider the following simultaneous equation

15342

yx

yx


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CRAMERS RULE FOR MATRIX SOLUTION

Cramers rule provides a simplified method of solving a system of linear equations through

the use of determinants. Cramers rules states that,

A

A

x i

i

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