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Finite mathematics topic summary: matrix algebra 
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. Move the mouse over the entries to see their names.


Operations with Matrices
Transpose
Sum, Difference
Scalar Multiple
Product

Examples
Transpose
Sum & Scalar Multiple
Product
Visit the Matrix Algebra Tool for online matrix algebra computations. Also visit the tambien el Tutorial on matrix algebra to see a more detailed discussion of these operations. 

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:
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. 
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


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. 