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.

Following is a 4×5 matrix. Move the mouse over the entries to see their names.

A =		0	1	2	0	3
		1/3	-1	10	1/3	2
		3	1	0	1	-3
		2	1	0	0	1

Transpose
The transpose, A^T, of a matrix A is the matrix obtained from A by writing its rows as columns. If A is an m×n matrix and B = A^T, then B is the n×m matrix with b_ij = a_ji.

Sum, Difference
If A and B have the same dimensions, then their sum, A+B, is obtained by adding corresponding entries. In symbols, (A+B)_ij = A_ij + B_ij. If A and B have the same dimensions, then their difference, A - B, is obtained by subtracting corresponding entries. In symbols, (A-B)_ij = A_ij - B_ij.

Scalar Multiple
If A is a matrix and c is a number (called a scalar in this context), then the scalar multiple, cA, is obtained by multiplying every entry in A by c. In symbols, (cA)_ij = c(A_ij).

Product
If A has dimensions m×n and B has dimensions n×p, then the product AB is defined, and has dimensions m×p. The entry (AB)_ij is obtained by multiplying row i of A by column j of B, which is done by multiplying corresponding entries together and then adding the results.

Transpose

0 1 2 T 1/3 -1 10

=

0 1/3 1 -1 2 10

Sum & Scalar Multiple

0 1 1/3 -1

+

2

1 -1 2/3 -2

=

2 -1 5/3 -5

Product

0 1 1/3 -1

1 -1 2/3 -2

=

2/3 -2 -1/3 5/3

Visit the Matrix Algebra Tool for on-line matrix algebra computations. Also visit the tambien el Tutorial on matrix algebra to see a more detailed discussion of these operations.

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 a matrix O all of whose entries are zero.

The various matrix operations, addition, subtraction, scalar multiplication and matrix multiplication, have the following properties.

The one rule that is conspicuously absent from this list is commutativity of the matrix product. In general, matrix multiplication is not commutative: AB is not equal to BA in general.

Following is the 4×4 identity matrix.

I =		1	0	0	0
		0	1	0	0
		0	0	1	0
		0	0	0	1

The following illustrates the failure of the commutative law for matrix multiplication.

A = 0 1 1/3 -1

B =

1 -1 2/3 -2

AB =	2/3 -2 -1/3 5/3
BA =	-1/3 2 -2/3 8/3

An important application of matrix multiplication is this: The system of linear equations

a11x1 + a12x2 + a13x3 + . . . + a1nxn	=	b1
a21x1 + a22x2 + a23x3 + . . . + a2nxn	=	b2
. . . . . . . . . . . . . .
am1x1 + am2x2 + am3x3 + . . . + amnxn	=	bm

can be rewritten as the matrix equation

AX = B

where

A =		a11	a12	a13	. . .	a1n
		a21	a22	a23	. . .	a2n
				. . . .	. . .
		am1	am2	am3	. . .	amn

X = [x₁, x₂, x₃, . . . , x_n]^T

B = [b₁, b₂, x₃, . . . , b_m]^T

The system

x	+	y	-	z	=	4
3x	+	y	-	z	=	6
x	+	y	-	2z	=	4
3x	+	2y	-	z	=	9

has matrix form

	1	1	-1			x		=		4		.
	3	1	-1			y				6
	1	1	-2			z				4
	3	2	-1							9

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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

AA^-1 = A^-1A = I.

When A is invertible we can solve the equation

AX = B

X = A^-1B.

The system of equations

	1	2	4			x		=		1
	2	4	6			y				1
	4	6	8			z				-1

has solution

	x		=		1	2	4		1		1
	y				2	4	6				1
	z				4	6	8				-1
			=		1	-2	1				1
					-2	2	-1/2				1
					1	-1/2	0				-1
			=		-2	.
					1/2
					1/2

In order to determine whether an n×n matrix A is invertible or not, and to find A¹ 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.

The matrix

A =		1	2	4
		2	4	6
		4	6	8

is invertible. The matrix

B =		1	2	4
		2	4	6
		2	4	7

is not.

The 2×2 matrix

A =		a	b
		c	d

A1 =	1 ad - bc		d	-b		.
			-c	a

	1	2		1	=	1 (1)(4) - (2)(3)		4	-2
	3	4						-3	1

=		-2	1		.
		3/2	-1/2

In a two-person zero sum game, each of the two players is given a choice between several prescribed moves at each turn, and each player's loss is equal to the other player's gain.

The payoff matrix of a two-person zero sum game has rows labeled by the "row player's" moves and columns labeled by the opposing "column player's" moves. The ij entry of the matrix is the payoff that accrues to the row player in the event that the row player uses move i and the column player uses move j.

Paper, Scissors, Rock
Rock beats (crushes) scissors; scissors beat (cut) paper, and paper beats (wraps) rock.
Each +1 entry indicates a win for the row player, -1 indicates a loss, and 0 indicates a tie.

Do you want to play? Click on a row strategy...

		Column Player
Row Player			0	-1	1
			1	0	-1
			-1	1	0

Click on a row move.

A player uses a pure strategy if he or she uses the same move at each round of the game. A mixed strategy is a method of playing a game where, at each round of the game, the player chooses a move at random so that each is used a predetermined fraction of the time.

We represent a mixed (or pure) strategy for the row player by a row matrix (probability vector)

R = [a   b   c  . . . ]

A mixed strategy for the column player is represented by a similar column matrix C. 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 R and C is given by

e = RPC

Here is a variant of "paper, scissors, rock in which "paper/paper" and "rock/rock" is no longer a draw.

		2	-1	1
		1	0	-1
		-1	1	-2

Suppose the row player uses the mixed strategy

R = [0.75   0   0.25]

C =		0
		0.5
		0.5		.

e	= RPC
	= [0.75   0   0.25]		2	-1	1			0
			1	0	-1			0.5
			-1	1	-2			0.5
	= -0.125

Minimax Criterion
A player using the minimax criterion chooses a strategy that, among all possible strategies, minimizes the effect of the other player's best counter-strategy. That is, an optimal (best) strategy according to the minimax criterion is one that minimizes the maximum damage the opponent can cause.

Finding the minimax strategy is called solving the game. In the third part of the tutorial for this topic we show a graphical method for solving 2×2 games. For general games, one uses the simplex method (see the next topic 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:

1. Each player makes the best possible move.
2. Each player knows that his or her opponent is also making the best possible move

Consider the following game.

		Column Strategy
			A	B	C
Row Strategy	1		0	-1	1
	2		0	0	2
	3		-1	-2	3

If the row player follows Principle 1, (s)he should never play Move 1 because Move 2 gives better payoffs no matter what move the column player chooses. (The payoffs in Row 2 are all at least as high as the corresponding ones in Row 1.)

Further, following Principle 2, the row player expects that the column player will never play Move A, since Move 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 corresponding ones in Column A.)

One move dominates another if all its payoffs are at least as advantageous to the player than the corresponding ones in the other move. In terms of the payoff matrix, we can say it this way:

1. Row r in the payoff matrix dominates row s if each payoff in row r is ≥ the corresponding payoff in row s.
2. Column r in the payoff matrix dominates column s if each payoff in row r is ≤ the corresponding payoff in column s.

Following the first principle of game theory, the move corresponding to a strictly dominated row or column will never be played, and both players are aware of this by the second principle. Thus each player following the principles of game theory will repeatedly eliminate dominated rows and columns as the case may be. (If two rows or columns are equal, then there is no reason to choose one over the other, so either may be eliminated.) This process is called reduction by dominance.

Consider the above game once again.

		Column Strategy
			A	B	C
Row Strategy	1		0	-1	1
	2		0	0	2
	3		-1	-2	3

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.

Reducing the Above Game by Dominance

Since Row 2 dominates Row 1 we eliminate Row 1 to obtain

		A	B	C
2		0	0	2
3		-1	-2	3

Since Column B now dominates both Columns A and C we eliminate both Columns A and C to obtain

		B
2		0
3		-2

Since the top row now dominates the bottom row, we eliminate the bottom row, and we are reduced to the following 1×1 matrix

In this case, we have solved the game by reduction by dominance: The row player should always play 2 and the column player should always play B. Since the corresponding payoff is 0, we say that the game is fair (neither player has an advantage over the other).

Note that we were lucky here: Not all games can be reduced by dominance to a 1×1 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.

1. All saddle points in a game have the same payoff value.
2. Choosing the row and column through any saddle point gives minimax strategies for both players. In other words, the game is solved via the use of these (pure) strategies.

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.

In the above game, there are two saddle points, shown in color.

		A	B	C
1		0	-1	1
2		0	0	2
3		-1	-2	3

Since the saddle point entries are zero, this is a fair game.

An input-output 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

(I - A)X = D,

X = (I - A)^-1D.

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.