Matrix for Engineers and mathematicians by NAGMA IRFAN

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

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• 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.

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Identity Matrix(Unit)
Square matrix with ones on the diagonal and zeros
elsewhere.
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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.
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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
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Transpose Matrix
Rows become columns and columns become rows
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Orthogonal matrix

• A matrix A is called orthogonal if AAT ATA
  I, i.e., AT A-1

Example prove that
is orthogonal.
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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
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Matrix addition is commutative
Matrix addition is associative
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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.
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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

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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???!!!!
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THANK YOU FOR YOUR KIND ATTENTION