Matrices: Basic Definitions

Matrices: Basic Definitions and Operations

A matrix is an array of entries; in our case these entries wll be numbers.

Matrices come in all possible rectangular shapes, the following are a number of examples of matrices:

In general, we denote a matrix by

Each a_ij is called an element of the matrix (or an entry of the matrix); this denotes the element in row i and column j. The entries of the matrix are organized in horizontal rows and vertical columns.

The size, or dimension, of the matrix is n x m, where n is the number of rows of the matrix, and m is the number of column of the matrix. For example, the matrices above are of dimensions 1x4, 3x1, 2x3, and 4x2 respectively.

A special kind of matrix is a square matrix, i.e. a matrix with the same number of rows and columns. If a square matrix has n rows and n columns, we say that the matrix has order n. Square matrices also have a special set of entries: those on the diagonal from top left to bottom right. This diagonal is called the principal, or main, diagonal, and its elements are called the principal, or main, diagonal elements.

Example The matrix is a square matrix of order 3, and its main diagonal elements are 1, -1, and 5.

The Algebra of Matrices

In some circumstances, it is possible to carry out arithmetic operations with matrices; we describe these here.

Scalar Multiplication To multiply a matrix A by a number c, multiply each entry of A by c. The result is a matrix of the same dimensions as those of A.
Example

Addition (and subtraction) If 2 matrices have the same dimensions, we may add them; we do so by adding elements in corresoponding places. In this case the result is a matrix of the same dimension as that of the original matrices.
Example

Note that matrix subtraction is not actually a new operation (just as subtraction of real numbers is not a separate operation) - to subtract matrix A from matrix B, simply multiply matrix B by -1 (using the scalar multiplication described above) and add them: A - B = A + (-1)B In this example, this would mean

Matrix Multiplication To multiply 2 matrices, they need to be of appropriate dimensions - but not necessarily the same dimensions! In fact, the first (left-hand) matrix should have the same number of columns as the second (right-hand) matrix has rows. So for example, one could multiply a 3x4 matrix by a 4x1, matrix, but not a 3x4 matrix by a 1x4. The resulting matrix will have as many rows as the first matrix, and as many columns as the second. So if we multiply a 3x4 matrix by a 4x1 matrix, the product will be a matrix of dimension 3x1.

Go through the next activity to see how to multiply matrices that can be multiplied one by the other.

And now practice multiplying matrices; if the multiplication CAN be carried out, decide what the dimension of the resulting product matrix is, and enter the number of rows and columns, then click "can", and continue by entering the entries for the product. If you have a problem, click "reveal" and you will get guidance. If you determine that the product CANNOT be carried out, click "can't".

Applications to Systems of Linear Equations

A major application of matrices is to the solution of systems of linear equations. The first order of business in this connection is the definition of appropriate matrices from the system:

Consider the system

2x - 3y + z = -1
3x + y + 2z = 0
x - 5y = 2

There are 2 matrices connected with the system:

The coefficient matrix, which is composed of the coefficients of the variables. The columns of the matrix are each the list of coefficients of one of the variables, and each row consists of the coefficients in one of the equations. In this case, we get the matrix . Note that this is a square matrix since there are 3 equations in 3 unknowns. There could, of course, be any size matrix here - a system of n equations in m unknowns will yield a matrix of dimension n x m.
The augmented matrix, which is the coefficient matrix with an addtional column "glued on" to the right; this column consists of the numbers to the right of the equations. In this example, we get the matrix .