Matrix Algebra

Matrix Algebra

BIBLIOGRAPHY

Real numbers can be used to convey one-dimensional information, such as a family’s total expenditure in a month. However, if one wants to record the monthly expenditures of two families (indexed by 1, 2) on three items—food, entertainment, and health (indexed by 1, 2, 3)—then one needs to use a rectangular array of real numbers, or a matrix. A matrix (A) is defined as a rectangular array of numbers, parameters, or variables. The members of the array are referred to as the elements of the matrix and are usually enclosed in brackets, parentheses, or double vertical lines.

The first row of matrix A provides information on the first family’s expenditures on food, entertainment, and health, and the second row gives similar information related to the second family. For instance, a₂₃ is the expenditure of the second family on health, and a₁₂ is the first family’s expenditure on entertainment. Matrix is a concept of linear algebra, and it has wide applications in many fields, including economics, statistics, computer programming, operations research, industrial organization, and engineering.

Like numbers, elementary operations such as addition and multiplication can also be performed on matrices. A typical element in matrix A is written as a_ij, which is the element located in row i and column j of the matrix. For instance, a ₁₃ is the element in the first row and third column, or the first family’s expenditure on health in the first month. Matrix A is also written as {a_ij } _2×3, which simply means that A is a matrix with two rows and three columns whose typical element is a_ij The expenditures of the two families in the second month on food, entertainment, and health can be written as another matrix, B = {b_ij } _2×3. Then, the sum of two months’ expenditures for both families on all three items can be written in the form of a third matrix C, which is the sum of the corresponding elements of A and B; it is defined as:

C= {a_ij + b_ij }_2×3 = A + B

Let us now assume that in the third month the two families spend exactly the same amount of money on every item as they spent in the first month. In other words, the matrix of expenditure in the third month is the same as the matrix of expenditure in the first month. Then we can write a matrix D, which will tell us the total expenditure of the two families in three months on each of the three items as:

D ={2a_ij + b_ij }_2×3 =2A + B.

2A is called scalar multiplication of a matrix, in which every element of A is multiplied by the scalar 2.

Multiplication of two matrices can be illustrated from input-output models used to illustrate producer theory in economics. Let us suppose that there are two firms, each producing three goods (indexed by 1, 2, 3) with two factors of production (indexed by 1, 2). Let A = {a_ij }_2×3 be the matrix of input-output coefficients. Thus, a_ij is the amount of factor i needed to produce one unit of good j. The first firm wants to produce X₁, X₂ X₃ and the second firm wants to produce Y₁, Y₂, Y₃ quantities of the three goods. Let us define a matrix of these outputs as:

Then the matrix multiplication of A and Z can be written as:

It may be noticed that A is a 2×3 and Z is a 3×2 matrix. Their product will be a 2×2 matrix. The general rule is that if A is m×n and Z is nXq then their product will be a matrix of the order of m×q. The number of columns of A must be equal to the number of rows of Z. If this condition is not satisfied, matrix multiplication is not possible. The meaning of the product matrix (AZ) is very simple. Suppose labor and capital are the two factors of production. The first column of AZ gives the total quantities of labor and capital used by the first firm to produce all three goods, and the second column of AZ gives the amounts of all factors used by the second firm to produce these three goods.

The transpose of a matrix A^T is obtained by interchanging the rows and columns of A. The i th row of A is the i th column of A^T. For instance, the transpose of A, written as A^T, is:

A square matrix is a matrix with an equal number of rows and columns. The equivalent of the number one is an identity matrix I, which is a square matrix with 1 in its principle diagonal and 0 elsewhere. The following identity matrix is of dimension 3×3:

Obviously, I^T = I, AI = IA = A, for any matrix A for which these products are defined. A null matrix (written as O) is a matrix whose elements are all zero.

Matrix operations satisfy certain mathematical laws:

1. A + (B+C) = (A+B) + C (associative law for addition)
2. A(BC) = (AB)C (associative law for multiplication)
3. A + B = B + A (commutative law)
4. c(A+B) = cA + cB, where c is a scalar (distributive law for scalar products)
5. C(A+B) = CA + CB, provided the products are defined (distributive law for matrix multiplication)

A crucial application of matrices is that matrix algebra can be used to solve a system of linear simultaneous equations of the form: AX = B. A is a matrix of order m×n whose elements are real numbers, X is an n × 1 matrix of variables whose values have to be solved, and B is an m × 1 matrix of right-hand-side constant terms of m linear equations. An example of these equations for the case m = n = 3 follows:

X + Y + Z = 4

3X + Y–Z = 6

X + Y – 2Z +4

In the above system of equations,

Something that is equivalent to division for real numbers is called matrix inversion in matrix algebra. Only a square matrix can be inverted. If A is a square matrix of order n×n, then its inverse, A^-1, is a matrix such that A^–1 = A^–1 A = I . If there is a system of linear simultaneous equations AX = B, where A is n×n, X is n× 1, and B is n× 1, then the solution is X = A^-1 B, provided A^-1 exists.

SEE ALSO Cholesky Decomposition; Hessian Matrix; Input-Output Matrix; Inverse Matrix; Jacobian Matrix; Linear Systems; Mathematical Economics; Programming, Linear and Nonlinear; Vectors

BIBLIOGRAPHY

Chiang, Alpha C., and Kevin Wainwright. 2005. Fundamental Methods of Mathematical Economics. 4th ed. Boston: McGraw Hill/Irwin.

Monica Das