Basic Definitions
An m×n matrix A is a rectangular array of real numbers with m rows and n columns. (Rows are horizontal and columns are vertical.) The numbers m and n are the dimensions of A. The real numbers in the matrix are called its entries. The entry in row i and column j is called a_{ij} or A_{ij}.

Example
Following is a 4×5 matrix with the entry A_{23} highlighted.


Operations with Matrices
Transpose
Sum, Difference
Scalar Multiple
Product

Examples
Transpose
Sum & Scalar Multiple
Product
Visit our Matrix Algebra Tool for online matrix algebra computations. 

Algebra of Matrices
The n×n identity matrix is the matrix I that has 1's down the main diagonal and 0's everywhere else. In symbols, I_{ij} = 1 if i = j and 0 if i j. A zero matrix is one whose entries are all 0. The various matrix operations, addition, subtraction, scalar multiplication and matrix multiplication, have the following properties.

Examples
Following is the 4×4 identity matrix.
The following illustrates the failure of the commutative law for matrix multiplication.


Matrix Form of a System of Linear Equations
An important application of matrix multiplication is this: The system of linear equations
can be rewritten as the matrix equation
where

Example
The system
has matrix form


Matrix Inverse
If A is a square matrix, one that has the same number of rows and columns, it is sometimes possible to take a matrix equation such as AX = B and solve for X by "dividing by A." Precisely, a square matrix A may have an inverse, written A^{1}, with the property that
When A is invertible we can solve the equation

Example
The system of equations
has solution


Determining Whether a Matrix is Invertible
In order to determine whether an n×n matrix A is invertible or not, and to find A^{1} if it does exist, write down the n×(2n) matrix [A  I] (this is A with the n×n identity matrix set next to it). Row reduce this matrix. If the reduced form is [I  B] (i.e., has the identity matrix in the left part), then A is invertible and B = A^{1}. If you cannot obtain I in the left part, then A is singular 
Examples
The matrix
is invertible. The matrix
is not. 

Inverse of a 2×2 Matrix
The 2×2 matrix

Example


TwoPerson Zero Sum Game
In a twoperson zero sum game, each of the two players is given a choice between several prescribed strategies at each turn, and each player's loss is equal to the other player's gain. The payoff matrix of a twoperson zero sum game has rows labeled by the row player's strategies and columns labeled by the column player's strategies. The ij entry of the matrix is the payoff that accrues to the row player in the event that the row player uses strategy i and the column player uses strategy j.

Example
Paper, Scissors, Rock
Do you want to play? Click on a row strategy...


Mixed Strategy, Expected Value
A player uses a pure strategy if he or she uses the same strategy at each round of the game. A mixed strategy is a method of playing a game where the rows or columns are played at random so that each is used a given fraction of the time. We represent a mixed (or pure) strategy for the row player by a row matrix (probability vector) S = [a b c . . . ]with the same number of entries as there are rows, where each entry represents the fraction of times the corresponding row is played (or the probability of using that strategy) and where a + b + . . . = 1. A mixed strategy for the column player is represented by a similar column matrix T. For both row and column players, pure strategies is represented by vectors in with a single 1 and the rest zeros. The expected value of the game with payoff matrix P corresponding to the mixed strategies S and T is given by e = SPTThe expected value of the game is the average payoff per round if each player uses the associated mixed strategies for a large number of rounds. 
Example
Here is a variant of "paper, scissors, rock in which "paper/paper" and "rock/rock" is no longer a draw.
Suppose the row player uses the mixed strategy S = [0.75 0 0.25](play paper 75% of the time, scissors 0% of the time and rock 25% of the time) and thje column player plays scissors and rock each 50% of the time; Then the expected value of the game is


Minimax Criterion, Fundamental Principles of Game Theory
Minimax Criterion
Finding the minimax strategy is called solving the game. In the texbook we show a graphical method for solving 2×2 games. For general games, one uses the simplex method (see the Chapter 4 Summary). However, one can frequently simplify a game and sometimes solve it by "reducing by dominance" and/or checking whether it is "strictly determined" (see below). Fundamental Principles of Game Theory When analyzing any game, we make the following assumptions about both players:

Example
Consider the following game.
If the row player follows Principle 1, (s)he should never play Strategy 1 since Strategy 2 gives better payoffs no matter what strategy the column player chooses. (The payoffs in Row 2 are all at least as high as the coresponding ones in Row 1.) Further, following Principle 2, the row player expects that the column player will never play Strategy A, since Strategy B gives better payoffs as far as the column player is concerned. (The payoffs in Column B are all at least as low as the coresponding ones in Column A.) 

Reducing by Dominance
One pure strategy dominates another if all its payoffs are more advantageous to the player than the corresponding ones in the other strategy. In terms of the payoff matrix, we can say it this way:

Example
Consider the above game once again.
Since the entries in Row 2 are ≥ the corresponding ones in Row 1, Row 2 dominates Row 1. Since the entries in Column B are ≤ the corresponding ones in Column A, Column B dominates Column A. 

Saddle Point, Strictly Determined Game
A saddle point is a payoff that is simultaneously a row minimum and a column maximum. To locate saddle points, circle the row minima and box the column maxima. The saddle points are those entries that are both circled and boxed. A game is strictly determined if it has at least one saddle point. The following statements are true about strictly determined games.
The value of a strictly determined game is the value of the saddle point entry. A fair game has value of zero, otherwise it is unfair or biased. 
Example
Row Player:In the above game, there are two saddle points, shown in color.
Since the saddle point entries are zero, this is a fair game. Our online game theory utility can be used to check any game (up to 55) for saddle points. Try it out. 

InputOutput Economic Models
An inputoutput matrix for an economy gives, as its jth column, the amounts (in dollars or other appropriate currency) of outputs of each sector used as input by sector j (for one year or other appropriate period of time). It also gives the total production of each sector of the economy for a year (called the production vector when written as a column). The technology matrix is the matrix obtained by dividing each column by the total production of the corresponding sector. Its ijth entry, the ijth technology coefficient, gives the input from sector i necessary to produce one unit of output from sector j. A demand vector is a column vector giving the total demand from outside the economy for the products of each sector. If A is the technology matrix, X is the production vector, and D is the demand vector, then
These same equations hold if D is a vector representing change in demand, and X is a vector representing change in production. The entries in a column of (I  A)^{1} represent the change in production in each sector necessary to meet a unit change of demand in the sector corresponding to that column, taking into account all direct and indirect effects. 