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Title: Matrix for Engineers and mathematicians by NAGMA IRFAN


1
MatricesNagma IrfanFor Engineers
2
1.1 Matrices
1.2 Operations of matrices
1.3 Types of matrices
1.4 Properties of matrices
1.5 Inverse of a matrix
3
Matrix - 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.
4
Examples Find the dimensions of each matrix.
Dimensions 3x2
Dimensions 4x1
5
How 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.

7
Identity Matrix(Unit)
Square matrix with ones on the diagonal and zeros
elsewhere.
8
Equal Matrices - two matrices that have the same
dimensions and each element of one matrix is
equal to the corresponding element of the other
matrix.
9
The 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
10
Transpose Matrix
Rows become columns and columns become rows
11
Orthogonal matrix
  • A matrix A is called orthogonal if AAT ATA
    I, i.e., AT A-1

Example prove that
is orthogonal.
12
1.2 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
13
Matrix addition is commutative
Matrix addition is associative
14
To 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.
15
1.4 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

16
1.5 Inverse of a 3?3 matrix
Cofactor matrix of
The cofactor for each element of matrix A
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Cofactor matrix of is then
given by
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Inverse matrix of is given by
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Questions???!!!!
20
THANK YOU FOR YOUR KIND ATTENTION
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