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MATRICES

This document provides an overview of matrices and matrix operations. It begins by stating the objectives of understanding matrix characteristics, applying basic matrix operations, knowing inverse matrices up to 3x3, and solving simultaneous linear equations up to 3 variables. It then defines what a matrix is, discusses matrix dimensions and types of matrices. The document outlines various matrix operations including addition, subtraction, multiplication and scalar multiplication. It provides examples of how to perform these operations. It also covers the transpose of a matrix and inverse matrices.

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BA301 ENGINEERING MATHEMATICS 3
INDEX OBJECTIVES DIMENSIONS TYPES OF MATRICES OPERATION ON MATRICES - Addition - Subtraction - Multiplication - Scalar Multiplication SOLVING MATRICES PROBLEMS TRANSPOSE OF MATRIX WHAT IS A MATRIX TUTORIAL A SOLVING SIMULTANEOUS EQUATION INVERSE MATRIX
OBJECTIVES: - Know characteristics of matrices - Apply basic operations on matrices - Know the inverse matrices (up to 3X3) - Solve simultaneous linear equations up to 3 variables
OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables Matrix : A rectangular arrangement of numbers in rows and columns Order/Dimensions/Size of a matrix is the number of the rows and columns Entries /Elements are the numbers in the matrix
OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables row       2 3 4 0 1 2 column (# no. of row) x ( # no. of column) DIMENSIONS OF THE MATRIX : So…….. This is a 2 X 3 matrix
OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables A matrix with m rows and n columns is called a matrix with dimensions m x n Example : Find the dimensions of the matrix.       1 9 7 3 2 4 ) a           2 9 3 7 0 5 ) b   3 8 1 )  c           5 1 2 3 4 8 10 9 7 ) d 2 X 3 3 X 2 1 X 3 3 X 3
OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables Find the dimensions of the following matrix   5 ) a       7 1 ) b       2 3 4 9 ) c 1 X 1 2 X 1 2 X 2
OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
No of rows = No of column        2 3 4 5 T 2 X 2 Square matrix            5 3 4 0 2 6 1 4 8 A 3 X 3 Square matrix OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
All entries = 0 EXCEPT at least one Of the diagonal entries            1 0 0 0 5 0 0 0 3 T diago nal            0 0 1 0 5 0 3 0 0 T Diagon al Matrix OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Diagonal entries = 1 All other entries = 0            1 0 0 0 1 0 0 0 1 I            0 0 1 0 1 0 1 0 0 I        1 0 0 1 I        0 1 1 0 I exampl e OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Element in row change into column and element in column change into rows            1 8 1 6 5 7 4 2 3 B            1 6 4 8 5 2 1 7 3 T B OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
-Both matrix must have the same dimensions to do addition operation -To add matrices, we add the corresponding elements OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Example :        4 5 1 2 A        8 7 6 5 B Find A + B         B A 2 + 5 1 + 6 5 + 7 4+ 8         12 12 7 7 B A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables CALC GEO
Find A + B     4 5 3 2 )     B and A a                     3 2 3 9 4 2 9 8 7 2 3 8 ) B and A b   7 7    B A            12 10 4 11 7 6 B A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
-Both matrix must have the same dimensions to do subtraction operation -To subtract matrices, we subtract the corresponding elements OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Example :        4 5 1 2 A        8 7 6 5 B Find A - B         B A 2 - 5 1 - 6 5 - 7 4- 8             4 2 5 3 B A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables GEO
Find A - B     4 5 3 2 )     B and A a                     3 2 3 9 4 2 9 8 7 2 3 8 ) B and A b   1 3    B A            6 6 10 7 1 10 B A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
          2 9 7 6 1 3 4 5 2           k k k k k k k k k 2 9 7 6 1 3 4 5 2 Multiply each element inside the matrix by OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Example :         6 3 4 2 5 Solve        5 X 2 5 X - 4 5 X - 3 5 X 6          30 15 20 10 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables GEO
Solve the following :   12 9 3 4 )  a   16 12             9 8 7 2 3 8 5 ) b           45 40 35 10 15 40 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Look at the dimension of matrices A and B to do MULTIPLICATIONS must be same size of product If the numbers of columns of A does NOT EQUAL the number of rows of B, then… A X B is UNDEFINED..!! OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Example :        7 4 3 1 5 2 A            8 3 2 B and Find A X B.                   8 3 2 7 4 3 1 5 2 B X A 2 X 3 3 X 1 sam e Size of product = 2 X 1 VERY IMPORTANT!! OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
       AXB (2 x 2) + (5x 3) + (1x 8) (3x2) + (4x3) + (7x8)            56 12 6 8 15 4 AXB        74 27 AXB                   8 3 2 7 4 3 1 5 2 B X A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables GEO
. . 12 6 4 5 7 8 2 5 1 2 3 9 ) AXB Find B and A a                        . . 7 5 6 4 5 8 ) PXQ Find Q and P b                        22 8 20 9 81 60 AXB   5   PXQ OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
. . 20 2 5 7 5 3 9 6 8 3 5 4 2 3 1 8 6 7 ) MXN Find N and M c                                       61 43 32 70 13 27 181 56 78 MXN                 2 8 5 3 1 4 ) U and T d        14 34 MXN OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Example : y and x Find x y x . 1 30 12 19 10 6 6 8 1 2 8 2 5 3 3                       3       9x-15 6 24 6y+3               1 30 12 19 10 6 6 8x         13 6 30 12 15 17 y x        1 30 12 19 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
17 x - 15 =19 17 x = 15 + 19 17 x = 34 x = 34/17 x = 2 6 y + 13 = 1 6 y = 13 - 1 6y = 12 y = 12/6 y = 2 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Tips : colum n row OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Example :        T A . . 2 5 1 2 3 9 T A Find A Given              9 2 5 -3 1 - 2 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables GEO
        12 6 4 5 7 8 ) B a Find transpose of the following matrices :             12 5 6 7 4 8 T B            6 7 8 ) P b   6 7 8  T P OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
C B d A B D c C A b B A a Find D C B A                                    ) ) ( ) ) ) : . 8 4 5 3 5 3 6 2 4 3 5 9 8 3 . 1 BA b AB a Find Find B and A If ) ) : : . 6 2 4 0 5 9 6 1 . 2                 . . 3 5 2 6 3 4 0 5 9 7 6 2 . 3 MN Find N and M If                       Answe r Answe r Answe r OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
2 ) ) : . 1 3 9 3 5 7 7 6 1 4 5 5 9 6 1 . 5 B b BA a Find B and A If                       T T B A b B A a Find B and A Given ) ( ) ) : . 2 4 8 8 6 9 3 4 5 5 6 2 . 6                     B A b B A a Find B and A matrix Given                         ) ) : . 3 9 4 8 6 1 3 0 5 9 6 2 . 4 Answe r Answe r Answe r OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
solution no C B d A B D c solution no C A b B A a Find D C B A                                                       ) 3 7 1 3 ) ( ) ) 1 7 12 6 ) : . 8 4 5 3 5 3 6 2 4 3 5 9 8 3 . 1 TUTORIAL A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
                                   42 56 20 36 ) 6 10 32 12 ) : : . 6 2 4 0 5 9 6 1 . 2 BA b AB a Find Find B and A If TUTORIAL A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
                                            41 43 55 . . 8 3 9 2 4 8 7 6 1 3 5 2 6 3 4 0 5 9 7 6 2 . 3 MN MN Find P and N M If TUTORIAL A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
                                               0 9 1 1 0 1 ) 6 9 9 17 12 3 ) : . 3 9 4 8 6 1 3 0 5 9 6 2 . 4 B A b B A a Find B and A matrix Given TUTORIAL A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
                                            73 72 39 67 76 69 32 57 106 ) 73 41 79 67 64 90 ) : . 1 3 9 3 5 7 7 6 1 4 5 5 9 6 1 . 5 2 B b BA a Find B and A If TUTORIAL A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
                                      5 9 14 12 11 11 ) 5 12 9 11 14 11 ) : . 2 4 8 8 6 9 3 4 5 5 6 2 . 6 B A b B A a Find B and A Given T T TUTORIAL A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
2 X 2 3 X 3 Formula : adjA A A    1 1 . . 6 5 4 5 1         A Find A ) 4 ( 5 ) 6 ( 5   A 20 30  A 10  A Example : 1. Determinant A GEO 2 X 2 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
2. Minor A        6 5 4 5 A        6 5 4 5 M What you can see?? 6 11  m 5 12  m 4 21  m 5 22  m        5 4 5 6 Minor OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
3. Cofactor A        5 4 5 6 Minor          5 4 5 6 Cofactor 4. Adjoint A          5 5 4 6 A Adj adjA A A    1 1            5 5 4 6 10 1 1 A 5. Inverse A                 2 1 2 1 5 2 5 3 10 1 GEO OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Formula : Example : . sin . 3 1 7 2 3 5 1 1 2 1 Matrix Inverse g u by A Find A              1. Determinant A              7 7 3 1 7 3 5 2 3 5 1 2 1 1 2 A   3 3 2     A   7 2 1     7 5 1     1 3 7      2 2 1     1 5 3     35 14 18     A   15 4 21    33  A 3 X 3 adjA A A    1 1 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
2. Minor             3 1 7 2 3 5 1 1 2 A     11 2 1 3 3 3 1 2 3 11 11 11             m m m     1 2 7 3 5 3 7 2 5 12 12 12           m m m     26 3 7 1 5 1 7 3 5 13 13 13             m m m     2 1 1 3 1 3 1 1 1 21 21 21           m m m     1 7 1 3 2 3 7 1 2 22 22 22            m m m     5 1 7 1 2 1 7 1 2 23 23 23            m m m     5 3 1 2 1 2 3 1 1 31 31 31             m m m     1 5 1 2 2 2 5 1 2 32 32 32            m m m     11 33 5 1 3 2 33 3 5 1 2 33              m m m OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
                 11 1 5 5 1 2 26 1 11 M 3. Cofactor                 11 1 5 5 1 2 26 1 11 K                 11 1 5 5 1 2 26 1 11 K 4. Adjoint                 11 5 26 1 1 1 5 2 11 A adj OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
5. Inverse A                  11 5 26 1 1 1 5 2 11 33 1 1 A                          3 1 33 5 33 26 33 1 33 1 33 1 33 5 33 2 3 1 33 1 1 A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
Cramer’s Rule Formula : A A z A A y A A x z y x    INVERSE MATRIX CRAMER’S RULE Inverse Matrix Formula : adjA A A    1 1 OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
CRAMER’S RULE Example : . ' sin , 8 4 8 2 5 9 2 6 4 7 5 Rule s Cramer g u by z and y x Find z y x z y x z y x                          8 2 4 8 2 2 6 9 2 6 1 5 7 1 5 A 1. Determinant A  A 4 2 5     2 9 1    8 6 7   7 2 2    5 9 8   1 6 4         ] 24 360 28 [ 336 18 40        A 2  A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
2. Determinant Ax                 8 8 4 8 2 5 9 2 1 4 7 1 x A 4 5 8 5 6 2  x A 4 2 4     8 9 1    8 5 7   7 2 8    4 9 8   1 5 4         ] 20 288 112 [ 280 72 32        x A 44  x A 3. Determinant Ay             8 2 4 2 5 6 9 6 4 5 7 5 x A 4 5 8 -1 -2 8  y A 4 5 5    2 9 4   8 6 7   7 5 2   5 9 8   4 6 4      ] 96 360 70 [ 336 72 100        y A 26   y A OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables
4. Determinant Az                8 2 8 8 2 2 6 5 2 6 1 5 4 1 5 z A  z A 8 2 5    8 6 4   2 5 1    4 2 2    5 5 8   1 6 8      ] 48 200 16 [ 192 10 80         z A 34   z A 22 2 44   x 13 2 26     y 17 2 34     z OBJECTIVES : -Know characteristics of matrices -Apply basic operations on matrices -Know the inverse matrices (up to 3X3) -Solve simultaneous linear equations up to 3 variables

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