5.matrix algebra.ppt
TRANSCRIPT
-
8/10/2019 5.MATRIX ALGEBRA.ppt
1/31
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.
1
-
8/10/2019 5.MATRIX ALGEBRA.ppt
2/31
The matrix [A] may also be denoted by
[A]m x n to show that [A] is a matrix with
mrow and ncolumn.
2
-
8/10/2019 5.MATRIX ALGEBRA.ppt
3/31
3
-
8/10/2019 5.MATRIX ALGEBRA.ppt
4/31
-
8/10/2019 5.MATRIX ALGEBRA.ppt
5/31
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
5
-
8/10/2019 5.MATRIX ALGEBRA.ppt
6/31
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
6
-
8/10/2019 5.MATRIX ALGEBRA.ppt
7/31
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.
7
-
8/10/2019 5.MATRIX ALGEBRA.ppt
8/31
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.
8
-
8/10/2019 5.MATRIX ALGEBRA.ppt
9/31
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.
9
-
8/10/2019 5.MATRIX ALGEBRA.ppt
10/31
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
10
-
8/10/2019 5.MATRIX ALGEBRA.ppt
11/31
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.
11
-
8/10/2019 5.MATRIX ALGEBRA.ppt
12/31
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]
12
-
8/10/2019 5.MATRIX ALGEBRA.ppt
13/31
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.
13
TRANSPOSE OF A MATRIX
-
8/10/2019 5.MATRIX ALGEBRA.ppt
14/31
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.
14
-
8/10/2019 5.MATRIX ALGEBRA.ppt
15/31
15
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
-
8/10/2019 5.MATRIX ALGEBRA.ppt
16/31
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].
16
PROPERTIES OF MATRIX ADDITION
-
8/10/2019 5.MATRIX ALGEBRA.ppt
17/31
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.
17
-
8/10/2019 5.MATRIX ALGEBRA.ppt
18/31
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.
18
-
8/10/2019 5.MATRIX ALGEBRA.ppt
19/31
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.
19
D t i th f ki l t i M d
-
8/10/2019 5.MATRIX ALGEBRA.ppt
20/31
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.
20
DAYS No.of.Car No.of.BusMonday 30 5
Tuesday 25 5
Wednesday 35 15
-
8/10/2019 5.MATRIX ALGEBRA.ppt
21/31
21
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
-
8/10/2019 5.MATRIX ALGEBRA.ppt
22/31
22
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
-
8/10/2019 5.MATRIX ALGEBRA.ppt
23/31
23
THIRD ORDER DETERMINAT
The determinant of a 3 x 3 matrix can be calculated as follows,
333231
232221
131211
aaa
aaa
aaa
A
-
8/10/2019 5.MATRIX ALGEBRA.ppt
24/31
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.
24
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.
-
8/10/2019 5.MATRIX ALGEBRA.ppt
25/31
25
i COFACTOR
-
8/10/2019 5.MATRIX ALGEBRA.ppt
26/31
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
-
8/10/2019 5.MATRIX ALGEBRA.ppt
27/31
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
27
-
8/10/2019 5.MATRIX ALGEBRA.ppt
28/31
28
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
-
8/10/2019 5.MATRIX ALGEBRA.ppt
29/31
29
AX = B
54
37A
2
1X
x
xand
29
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
-
8/10/2019 5.MATRIX ALGEBRA.ppt
30/31
30
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
-
8/10/2019 5.MATRIX ALGEBRA.ppt
31/31
31
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