Matrix algebra

             Matrices provide a means of storing large quantities of information in such a way that

each piece can be easily identified and manipulated. They permit the solution of large

systems of linear equations to be carried out in a logical and formal way so that computer

implementation follows naturally. Applications of matrices extend over many areas of

engineering including electrical network analysis and robotics.

1-BASIC DEFINITIONS

             Matrix is a rectangular pattern or array of numbers.

    For example: 

              are all matrices. Note that we usually use a capital letter to denote a matrix, and enclose

the array of numbers in brackets. To describe the size of a matrix we quote its number

of rows and columns in that order so, for example, an r × s matrix has r rows and s

columns. We say the matrix has order r × s.

              An r × s matrix has r rows and s columns.

Example: Describe the sizes of the matrices A, B andC at the start of this section, and give examples

of matrices of order 3 × 1, 3 × 2 and 4 × 2.

Solution:

                where ai j represents the number or element in the ith row and jth column. A matrix with

a single column can also be regarded as a column vector.The operations of addition, subtraction and 

multiplication are defined upon matricesand these are explained.

2- ADDITION, SUBTRACTION AND MULTIPLICATION

    2.1- Matrix addition and subtraction

        Two matrices can be added (or subtracted) if they have the same shape and size, that is the

same order. Their sum (or difference) is found by adding (or subtracting) corresponding

elements as the following example shows.

Example: 

Solution:

        Now a + e is exactly the same as e + a because addition of numbers is commutative.

        The same observation can be made of b + f , c + g and d + h. Hence C + D = D +C.

        The addition of these matrices is therefore commutative. This may seem an obvious statement but 

we shall shortly meet matrix multiplication which is not commutative, so in general commutativity 

should not be simply assumed.

                     Matrix addition is commutative, that is  A+B = B+A

              Matrix addition is associative, that is A+(B+C) = (A+B)+C

    2.2- Scalar multiplication

           Given any matrix A, we can multiply it by a number, that is a scalar, to form a new matrix of the same order as A. This multiplication is performed by multiplying every element of A by the number.

Example: 

Solution:

In general we have

    2.3- Matrix multiplication

            Matrix multiplication is defined in a special way which at first seems strange but is in fact very 

useful. If A is a p× q matrix and B is an r × s matrix we can form the productAB only if q = r; that is,  

only if the number of columns in A is the same as the numberof rows in B. The product is then a p × s 

matrix C, that is        C = AB         

                       where A = p×q

                                  B = q×s

                                  C = p×s

Example: 

Solution: