Introduction to matrix multiplication

Introduction to matrix multiplication

What is a matrix ?

A matrix is a rectangular array of numbers.

An item in a matrix is called an entry or an element.The horizontal lines in a matrix are called rows and the vertical lines are called  columns. A matrix with m rows and n  columns is called an m-by-n matrix (or m×n matrix)  and m and n are called its dimensions.

Example:


The size of the above matrix can be specified as 3 x 2 matrix as it has 3 rows and 2 columns.

Definition on Matrix Multiplication

Definition of matrix multiplication

Two matrices A and B are said to be compatible for matrix multiplication if the number of columns in A is equal to number of rows in B.If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product C is the m-by-p matrix whose entries are given by dot-product of the corresponding row of A and the corresponding column of B.

If A = (aij )m x n and B = (bjk )n x p are two matrices then their product is defined to be the matrix C = (cik )m x p

Where cik = `sum_(j=1)^n`aij bjk

Example:

               
          Now we need to multiply the rows of A by the columns of B. For this first take the first row of A and the first column of B, and we multiply the first entries, then the second entries, and then we add the two products. The sum is one entry in the product matrix AB; in fact, being the product of row 1 and column 1, the result is the 1,1-entry of AB. Then we continue in like manner. For instance, the sum of the products from row 2 of A and column 1 of B is the 2,1-entry of AB.

Here A is a 2 x 2 matrix, B is a 2 x 3 matrix and hence AB is a 2 x 3 matrix.
Properties on Matrix Multiplication

1. Associative property of matrix multiplication

                 A(BC) = (AB)C

2. Distributive property of matrix multiplication

                A(B + C) = AB + AC
                (A + B)C = AC + BC

3. Scalar Multiplication

               c(AB) = (cA)B = A(cB)

4. Muliplication of matrices is not Commutative

   AB  need not  to be equal to BA

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