Title: Matrix for Engineers and mathematicians by NAGMA IRFAN
1MatricesNagma IrfanFor Engineers
21.1 Matrices
1.2 Operations of matrices
1.3 Types of matrices
1.4 Properties of matrices
1.5 Inverse of a matrix
3Matrix - a set of mn no.s arranged in a
rectangular array having m rows n columns,the
no.s being enclosed by brackets or (). Element
- each value in a matrix either a number or a
constant. Dimension - number of rows by number of
columns of a matrix. A matrix is named by its
dimensions.
4Examples Find the dimensions of each matrix.
Dimensions 3x2
Dimensions 4x1
5How about solving
1.1 Matrices
Consider the following set of equations
It is easy to show that x 3 and y 4.
Matrices can help
6- 1.3 Types of Matrices
- Column Matrix - a matrix with only one column.
- Row Matrix - a matrix with only one row.
- Square Matrix - a matrix that has the same
number of rows and columns.
7Identity Matrix(Unit)
Square matrix with ones on the diagonal and zeros
elsewhere.
8Equal Matrices - two matrices that have the same
dimensions and each element of one matrix is
equal to the corresponding element of the other
matrix.
9The transpose of a matrix
- The matrix obtained by interchanging the rows and
columns of a matrix A is called the transpose of
A (write AT).
Example The transpose of A is
10Transpose Matrix
Rows become columns and columns become rows
11Orthogonal matrix
- A matrix A is called orthogonal if AAT ATA
I, i.e., AT A-1
Example prove that
is orthogonal.
121.2 Operations of matrices
Properties
- Matrices A, B and C are
- A(B C) AB AC
- (A B)C AC BC
- A(BC) (AB) C
- AB ? BA in general
- AB 0 NOT necessarily imply A 0 or B 0
- AB AC NOT necessarily imply B C
However
13Matrix addition is commutative
Matrix addition is associative
14To multiply matrices A and B look at their
dimensions
MUST BE SAME
SIZE OF PRODUCT
If the number of columns of A does not equal the
number of rows of B then the product AB is
undefined.
151.4 Properties of matrix
- (AB)-1 B-1A-1
- (AT)T A and (lA)T l AT
- (A B)T AT BT
- (AB)T BT AT
161.5 Inverse of a 3?3 matrix
Cofactor matrix of
The cofactor for each element of matrix A
17Cofactor matrix of is then
given by
18Inverse matrix of is given by
19Questions???!!!!
20 THANK YOU FOR YOUR KIND ATTENTION