Main Page Everything for Finite Math Everything for Applied Calc Everything Topic Summaries On Line Tutorials On Line Utilities               ← Previous Summary Next Summary → Tutorial for This Topic Review Exercises Textbook Lleveme a la Página Español Finite mathematics topic summary: matrix algebra Tools: Matrix Algebra Tool | Pivot and Gauss-Jordan Tool Subtopics: Basic Definitions | Operations with Matrices | Algebra of Matrices | Matrix Form of a System of Linear Equations | Matrix Inverse | Determining Whether a Matrix is Invertible | Inverse of a 2 × 2 Matrix | Application: Input-Output Economic Models

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 aij or Aij. Top of Page
Example

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

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Operations with Matrices

Transpose
The transpose, AT, 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 = AT, then B is the n×m matrix with bij = aji.

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 = Aij + Bij. If A and B have the same dimensions, then their difference, A - B, is obtained by subtracting corresponding entries. In symbols, (A-B)ij = Aij - Bij.

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(Aij).

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.

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Examples

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.

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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:

Iij = 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.

 A+(B+C) = (A+B)+C Additive associative law A+B = B+A Additive commutative law A+O = O+A = A Additive identity law A+( - A) = O = ( - A)+A Additive inverse law c(A+B) = cA+cB Distributive law (c+d)A = cA+dA Distributive law 1A = A Scalar unit 0A = O Scalar zero A(BC) = (AB)C Multiplicative associative law AI = IA = A Multiplicative identity law A(B+C) = AB + AC Distributive law (A+B)C = AC + BC Distributive law OA = AO = O Multiplication by zero matrix (A+B)T = AT + BT Transpose of a sum (cA)T = c(AT) Transpose of a scalar multiple (AB)T = BTAT Transpose of a matrix product

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.

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Examples

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

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Matrix Form of a System of Linear Equations

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, . . . , xn]T
and
B = [b1, b2, x3, . . . , bm]T

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Example

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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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

AA-1 = A-1A = I.
If A has an inverse we say that A is invertible, otherwise we say that A is singular.

When A is invertible we can solve the equation

AX = B
by multiplying both sides by A-1, which gives us
X = A-1B.

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Example

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

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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.

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Examples

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.

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Inverse of a 2×2 Matrix

The 2×2 matrix

 A = a b c d
is invertible if ad - bc is nonzero and is singular if ad - bc = 0. The number ad - bc is called the determinant of the matrix. When the matrix is invertible its inverse is given by the formula
 A 1 = 1 ad - bc d -b . -c a

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

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

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Input-Output Economic Models

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,
or
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.

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Last Updated: July 2007