Matrices

A matrix, singular for matrices, is a rectangular array of numbers. Matrices naturally arise in describing a special class of functions called linear transformations. But the concept of matrices originated in the work of the two mathematicians Arthur Cayley and James Sylvester while solving a system of linear equations. In 1857, Cayley wrote Memoir on the Theory of Matrices.

A matrix can be seen as a collection of rows of numbers. Each number is called an element, or entry, of the matrix. An illustrative example of a matrix, C, is below.

The order of the numbers within the row as well as the order of the rows are important. A matrix can be described by rows and columns. C has 3 rows and 3 columns, and hence it is a 3 × 3 matrix. A 2 × 3 matrix has 2 rows and 3 columns and a 4 × 2 matrix has 4 rows and 2 columns.

The size or dimension of a matrix is the number of rows and the number of columns, written in that order, and in the format m × n, read "m by n." If n = m, which means that the number of rows equals the number of columns, then the matrix is called a square matrix.

Symbolically, the elements of the first row of a matrix are a₁₁, a₁₂, a₁₃,… The second row is a₂₁, a₂₂, a₂₃, …, and so on. The first digit in the subscript indicates the row number and the second digit indicates the column number. Therefore, the element a_ij is located in row i and column j.

Addition and Subtraction

Addition and subtraction are defined for matrices. To add or subtract two matrices, they must have the same dimension. Two matrices are added or subtracted by adding or subtracting the corresponding element of each matrix. A matrix can also be multiplied by a real number. If C is a matrix and k a real number, then the matrix k C is formed by multiplying each element of C by k.

How can two matrices be multiplied? A useful definition of matrix multiplication involves a unique and unusual technique. To multiply two matrices A and B, the number of columns of the first matrix A has to equal the number of rows of the second matrix B. Let A be an m × n matrix, and B an n × p matrix. The product AB is a matrix C that has the dimension of m × p. The first element c₁₁ of the matrix C is obtained by multiplying the elements of the first row of A with the elements of the first column of B.

The second element in the first row, c₁₂, is obtained by multiplying the first row of A by the second column of B. Similarly, multiplying the row i of A with column j of B produces the element c_ij of matrix C.

Matrix multiplication is not commutative: that is, AB is not always equal to BA. The number 1 has a special property in arithmetic. Every number multiplied by 1 remains unchanged, hence 1 is called the multiplicative identity. Is there a special matrix which when multiplied by any matrix A leaves A unchanged? If A is a square matrix, then there exists an identity matrix I, such that AI = IA = A.

If A is a 3 × 3 square matrix, then I is given by determinant.

In an identity matrix, all elements are 0 except the diagonal elements that are all equal to 1. In arithemetic, a nonzero number has a multiplicative inverse, such that multiplying the number and its multiplicative inverse always produces 1, the identity. Does every square matrix B has an inverse B^-1? The answer is no. There is a special class of square matrices that have an inverse, such that B B⁻¹ = I = B⁻¹B. These square matrices have a nonzero determinant.

Matrices

Addition and Subtraction

Bibliography