Inteference

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Mathematics
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Session
Matrices and Determinants-1
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Session Objectives

• Matrix
• Types of Matrices
• Operations on Matrices
• Transpose of a Matrix
• Symmetric and Skew-symmetric Matrix
• Class Exercise

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Matrix
A matrix is a rectangular array of numbers, real
or complex.
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Order of a Matrix
A matrix with m rows and n columns has an order m
x n.
Examples
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Example - 1
A matrix has 16 elements, what is the possible
number of columns it can have.
Solution
The possible orders for the matrix are (1 x 16),
(2 x 8), (4 x 4), (8 x 2),(16 x 1) So, the
number of possible columns are 16, 8, 4, 2 and 1.
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Example-2
Solution
Here i can take the values 1 and 2 and j can
take the values 1, 2 and 3. Hence, the order of
the matrix is (2 x 3).
Now,
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Types of Matrices
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Types of Matrices
Zero matrix
Square matrix
Diagonal matrix
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Types of Matrices
Scalar matrix
Identity matrix
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Equality of Matrices
Two matrices A aij and B bij are
equal, if they have the same order and aij
bij for all i and j.
Example
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Addition of Matrices
Example
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Multiplication of a Matrix by a Scalar
Example
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Properties of Addition
If the order of the matrices A, B and C is same,
then (i) A B B A
(Commutativity) (ii) (A B) C A (B C)
(Associativity) (iii) If m and n are
scalars, then (a) m(A B) mA mB
(b) (m n)A mA nA
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Example - 3
Solution
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Example - 4
Solution A B C 0
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Multiplication of Matrices
Let A aijm x n be a m x n matrix and B
bijn x p be a n x p matrix , i.e. , the number
of columns of A is equal to the number of rows of
B. Then their product AB is of order m x p and is
given as
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Example
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Properties of Multiplication of Matrices
If both sides are defined, then (i) A(BC)
(AB)C (Associativity) (ii)
A ( B C ) AB AC and (A B) C AC
BC

( Multiplication is
distributive over addition)
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Example - 5
Solution
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Example - 6
Solution
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Example - 7
Solution
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Solution Contd.
Comparing the corresponding elements of the two
matrices , we get 3k-2 1, -2k -2 , 4 4k
, -4 -2k 2 Taking any of the four equations,
we get k1
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Example - 8
Solution
Hence , A2 12A IO
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Transpose of a Matrix
A matrix obtained by changing rows into columns
or columns into rows is called transpose of the
matrix ( say A ). If the matrix is A, then its
transpose is denoted as AT or A .

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Example - 9
Solution
Hence, (AB)TATBT
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Example - 10
Solution
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Example - 11
Solution
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Solution (Cont.)
Equating the elements of column 2 , we get
2y2 z2 0 (i)
Adding (ii) and (iii), we get
Form (i), z2 2y2
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Solution (Cont.)

• Putting the value of x2 and z3 in (ii), we get

Putting the value of y2 in (i), we get
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Example - 12
Solution
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Symmetric and Skew Symmetric Matrix
A square matrix A is called a symmetric matrix,
if AT A. A square matrix A is called a skew-
symmetric matrix, if AT - A. Any square matrix
can be expressed as the sum of a symmetric and a
skew- symmetric matrix.
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Example - 13
Solution
As AT - A, A is a skew symmetric matrix
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Example - 14
Solution
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Solution Cont.
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Solution Cont.
Therefore, P is symmetric and Q is skew-
symmetric . Further, PQ A Hence, A can be
expressed as the sum of a symmetric and a skew
-symmetric matrix.
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THANK YOU