Final matrix ppt

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Information about Final matrix ppt
Education

Published on March 10, 2014

Author: rohan1195

Source: slideshare.net

Description

Matrix ..types
addition
subtraction
multiplication
inverse

Algebra

421 0011 0010 1010 1101 0001 0100 1011 DEFINATION A a11 , , a1n a21 , , a2n am1 , , amn Aij

421 0011 0010 1010 1101 0001 0100 1011 TYPES VECTOR MATRIX SCALAR MATRIX SQUARE MATRIX SYMMETRIC MATRIX DIAGONAL MATRIX IDENTITY MATRIX

421 0011 0010 1010 1101 0001 0100 1011 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 1745B

421 0011 0010 1010 1101 0001 0100 1011 SCALAR MATRIX A diagonal matrix in which all of the diagonal elements are equal is called Scalar Matrix 500 050 005 B

421 0011 0010 1010 1101 0001 0100 1011 SQUARE MATRIX 312 163 745 A A matrix with same number of rows and columns is a square matrix 53 47 B

421 0011 0010 1010 1101 0001 0100 1011 SYMMETRIC MATRIX

421 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:- 500 050 005 B

421 0011 0010 1010 1101 0001 0100 1011 IDENTITY MATRIX An identity matrix is a diagonal matrix with 1 & only 1 on diagonal .The diagonal matrix is always denoted as I 100 010 001 A

421 0011 0010 1010 1101 0001 0100 1011 Addition Subtraction Multiplication Inverse Matrix Operations

421 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

421 0011 0010 1010 1101 0001 0100 1011 A a11 a12 a21 a22 B b11 b12 b21 b22 22222121 12121111 baba baba BA If and then 22222121 12121111 baba baba BA also

421 0011 0010 1010 1101 0001 0100 1011 EXAMPLE 1 4 2 3 5 8 6 7 + = 6 12 8 10 A B+ = C

421 0011 0010 1010 1101 0001 0100 1011 EXAMPLE 1 4 2 3 5 8 6 7 - = 4 4 4 4 B A- = C

421 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.

421 0011 0010 1010 1101 0001 0100 1011 Step I 1 4 2 3 5 8 6 7 x = A Bx =

421 0011 0010 1010 1101 0001 0100 1011 Step II 1 4 2 3 5 8 6 7 x = A Bx = C (5x1) C11 = A11 x B11k=1 n

421 0011 0010 1010 1101 0001 0100 1011 Step III 1 4 2 3 5 8 6 7 x = A Bx = C (5x1)+(6x3) C11 = A12 x B21k=2 n

421 0011 0010 1010 1101 0001 0100 1011 Step IV 1 4 2 3 5 8 6 7 x = A Bx = C 23 (5x2)+(6x4) C12 = A1k x Bk2k=1 n

421 0011 0010 1010 1101 0001 0100 1011 Step V 1 4 2 3 5 8 6 7 x = A Bx = C 23 (7x1)+(8x3) 34 C21 = A2k x Bk1k=1 n

421 0011 0010 1010 1101 0001 0100 1011 Step VI 1 4 2 3 5 8 6 7 x = A Bx = C 23 34 (7x2)+(8x4)31 C22 = A2k x Bk2k=1 n

421 0011 0010 1010 1101 0001 0100 1011 Result 1 4 2 3 5 8 6 7 x = A Bx = C 23 34 31 46 m x n n x p m x p

Inverse

421 0011 0010 1010 1101 0001 0100 1011 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

421 0011 0010 1010 1101 0001 0100 1011 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

421 0011 0010 1010 1101 0001 0100 1011 Step II--Swap elements a11 and a22 • Swap elements a11 and a22 Thus becomes 2 3 5 1 =A 3 2 5 1

421 0011 0010 1010 1101 0001 0100 1011 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

421 0011 0010 1010 1101 0001 0100 1011 Step IV • Divide every element by the determinant Thus becomes (no change as the determinant was 1) 3 2 -5 -1 =A 3 2 -5 -1

421 0011 0010 1010 1101 0001 0100 1011 Step V– Check the result • Check results with A -1 A = I Thus equals 3 2 -5 -1 x 1 1 0 0 2 3 5 1

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