Page 306 - AI Ver 3.0 Class 11
P. 306

Brainy Fact


                      We all know that computer understands data only in the form of binary/hexadecimal
                      numbers. Then, how are images processed by computers?
                      Consider, the given image:                                                1  5  0  3 0  0 3  3 0
                                   On the computer/mobile, the image is represented as a     3 2  0 6  2  4 0  3 6  1  4 0  1 2
                                   combination of pixels. This is represented mathematically as   2  4  1 1  5  0 3  3 2  3  0 1
                                                                                             3  0  2  4  2  1
                                   matrices!                                                 2  6  1  0  6  0  3  6
                                                                                             2  4  2  4  4  1  6  1
                                                                                             2  6  1  0  6
                                                                                             2  4



                      Operations on Matrices


              Addition of Matrices
              The sum of two matrices is a matrix obtained by adding the corresponding elements of the given matrices. Also, the
              two matrices have to be of the same order. For example:

                                                           2   4       1  0
                                                                       
                                                      A =      6  8     B = − 2  7   
                                                                       
                                                          7   5           4 
                                                                32     3     32
                                                                                ×
                                                                 ×
                                                            21   4 0   +    3  4 
                                                            +
                                                                             
                                                    A B =   6+−  8 7 ⇒     4  15 
                                                                    +
                                                      +
                                                             ( ) 2
                                                                      
                                                                          10 9 
                                                            +
                                                                    +
                                                            7 3  54          
              Difference of Matrices
              If A and B are two matrices of the same order, then the difference A – B is defined as a matrix where each element is
              obtained by subtracting the corresponding elements (a  – b ). For example:
                                                                    ij
                                                                ij
                                                         12 3        3  2  3−
                                                    A =         B =        
                                                         230   23    −  1 0 −  2   23
                                                                ×
                                                                                ×
                                                         13  2 −−      3 3        2 40−
                                                                         −
                                                        −
                                                                 ( ) 2
                                                 AB   −=                       ⇒        
                                                                          ( ) 2 
                                                       2 ( 1) 3 0
                                                       
                                                           − −  −      0 − −      3  32   23
                                                                                            ×
              Transpose of a Matrix
                                                                                                                      T
              A matrix obtained by interchanging the rows and columns of a matrix. Transpose of a matrix A is denoted by A’ or A .
              If order of matrix A is m × n, then order of transpose of matrix A, i.e., A’ is n x m. For example:
                                                                              14  
                                                        12 3                    
                                                                           ‘
                                                     A =         A OR A = 24     
                                                                    T
                                                                              
                                                         45 6  ×23           36 
                                                                                   ×32
              Multiplication of a Matrix by a Scalar
              A scalar is any number. So, if A is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying
              each element of A by the scalar k.
                    304     Touchpad Artificial Intelligence (Ver. 3.0)-XI
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