let  A be any square matrix

Then A can be expressed as

Adding and subtracting A’ on R.H.S

Hence Q is a skew-symmetric matrix

Hence A=P+Q

Hence every square matric can be expressed as a sum of a symmetric and skew-symmetric matrix 

Using crammer’s rule

By Matrix method

4

X-a

For a given  small we find a positive numbers  depending upon  such that

.  Statement If  

Differentiating (1)  partially  t’x’

Adding (2) and (3) we get

Dummy

Reduction(Row)

Column Reduction

Total cost =18+13+19=50

Game theory :

Game theory is a branch of mathematics concerned with the analysis of conflict situation It involves determining a strategy for a given situation and the costs or benefits realized by using the strategy.

Characteristics

1. There is a conflict between two or more players resulting in a win for some and a loss for others.

2. There is a way to start the game

3. During each move, single slayers are forced to make choices or the choices are assigned by random devices

 Two-person game :

A two-person game is characterized by the strategies of each player and the payoff matrix

The two-person game is also called zero-sum game because one player wins whatever the other player loses 

   ​​​​​​​

Let x and y be tur non-empty sets

Is said to be one-one if every let if x has a unique image in y.

Onto  

If every let f y has an ore image in x then

Liners f×n:

When A liners quation is represented in slope

Intercept form  

Can be regarded as a linear  

 Adjoint of A matrix

The transpose of the co factor matrix of matrix is adjoint of a matrix.

Limitations of graphic method

1. This method is  limited to lineal programming models containing only two decision variables
2. Factors such as uncertainty and time are not taken into consideration

An m n matrix which gives the passible outcome of a two person zero sum game when plays A has m passible moves and [layer B n moves the analysis of the aim of game theory the so called augment pay off matrix is defined as follows 

 Least cost entry method :  The least cost method is another method used to obtain the initial feasible soln for the transportation problem here the allocation begins with the cell which has the minimum cost the loves cost cells are chosen over the higher to have the least cost of transportation

Balanced assignment problem is assignment problem where number of facilities is qual to the numbers of jobs.