final matrix ppt
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Algebra
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DEFINATION
A matrix is a collection of numbers arranged into a fixed number of rows
and columns.
A
a11 ,, a1n
a21 ,, a2n
am1 ,, amn
Aij
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TYPES
VECTOR MATRIX
SCALAR MATRIX
SQUARE MATRIX
SYMMETRIC MATRIX
DIAGONAL MATRIX
IDENTITY MATRIX
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VECTOR MATRIX
A vector is a special type of matrix that has only one row (called a row vector) or one column (column vector ) .Below A is a column while B is a row vector.
3
1
4
A 1 7 4 5 B
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SCALAR MATRIX
A diagonal matrix in which all of the diagonal elements are equal is called Scalar Matrix
5 0 0
0 5 0
0 0 5
B
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SQUARE MATRIX
3 1 2
1 6 3
7 4 5
A
A matrix with same number of rows and columns is a square matrix
5 3
4 7 B
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SYMMETRIC MATRIX
A symmetric matrix is square matrix in which = for all i and j matrix A is symmetric matrix.
OR, A=A'
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0011 0010 1010 1101 0001 0100 1011
DIAGONAL MATRIX
A diagonal matrix is a matrix is a symmetric matrix where all the off diagonal elements are 0 .
Ex:-
5 0 0
0 5 0
0 0 5
B
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IDENTITY MATRIX
An identity matrix is a diagonal matrix with 1 & only 1 on diagonal .The diagonal matrix is always denoted as I
1 0 0
0 1 0
0 0 1
A
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Addition
Subtraction
Multiplication
Inverse
Matrix Operations
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0011 0010 1010 1101 0001 0100 1011
Two matrices can be added or subtracted if and only if the number of rows and columns are same.
ADDITION AND SUBTRACTION
4251 3
0011 0010 1010 1101 0001 0100 1011A
a11 a12
a21 a22
B b11 b12
b21 b22
22222121
12121111
baba
babaBA
If and
then
22222121
12121111
baba
babaBA
also
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EXAMPLE
1
4
2
3
5
8
6
7+ =
6
12
8
10
A B+ = C
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EXAMPLE
1
4
2
3
5
8
6
7- =
4
4
4
4
B A- = C
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0011 0010 1010 1101 0001 0100 1011
Multiplication Matrices A and B can be multiplied if the no. of coloum of first matrix is same as the no. of rows of the second
[r x c] and [s x d]
c = s
i.e.
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Step I
1
4
2
3
5
8
6
7x =
A Bx =
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Step II
1
4
2
3
5
8
6
7x =
A Bx = C
(5x1)
C 11 = A 11 x B 11k=1
n
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Step III
1
4
2
3
5
8
6
7x =
A Bx = C
(5x1)+ (6x3)
C 11 = A 12 x B 21k=2
n
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Step IV
1
4
2
3
5
8
6
7x =
A Bx = C
23 (5x2)+ (6x4)
C 12 = A 1k x B k2k=1
n
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Step V
1
4
2
3
5
8
6
7x =
A Bx = C
23
(7x1)+ (8x3)
34
C 21 = A 2k x B k1k=1
n
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Step VI
1
4
2
3
5
8
6
7x =
A Bx = C
23 34
(7x2)+ (8x4)31
C 22 = A 2k x B k2k=1
n
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0011 0010 1010 1101 0001 0100 1011
Result
1
4
2
3
5
8
6
7x =
A Bx = C
23 34
31 46
m x n n x p m x p
Inverse
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Find determinant
Swap the diagonal
elements(a11 and a22)
Change signs of non-
diagonal elements(a12
and a21)
Divide each element by determinant
INVERSE OF A 2x2 MATRIX
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Step I--Find the determinant
A determinant is a scalar number which is calculated from a matrix. This number can determine whether a set of linear equations are solvable, in other words whether the matrix can be inverted.
• Find the determinant = (a11 x a22) - (a21 x a12)
For
det(A) = (2x3) – (1x5) = 1
2
3
5
1=A
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Step II--Swap elements a11 and a22
• Swap elements a11 and a22
Thus
becomes
2
3
5
1=A
3
2
5
1
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Step III--Change sign of a12 and a21
• Change sign of a12 and a21
Thus
becomes
3
2
5
1=A
3
2
-5
-1
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0011 0010 1010 1101 0001 0100 1011
Step IV
• Divide every element by the determinantThus
becomes
(no change as the determinant was 1)
3
2
-5
-1=A
3
2
-5
-1
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Step V– Check the result
• Check results with A-1 A = IThus
equals
3
2
-5
-1x
1
1
0
0
2
3
5
1
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