An m x n ( "m by n" ) matrix is a rectangular array of numbers arranged in m rows (horizontal lines) and n columns (vertical lines).

Examples:

A matrix with 3 rows and 2 columns : a 3 x 2 matrix (read "a 3 by 2 matrix")

[2 3]

[-4 5]

[6 -8]

A matrix with 3 rows and 3 columns : a 3 x 3 matrix

[2 3 -4]

[4 5 6]

[6 0 8]

In general an m x n matrix A has the form

[a_{ 11} a_{ 12} ... a_{ 1n}]

[a_{ 21} a_{ 22} ... a_{ 2n}]

[. ]

A = [. ]

[. ]

[a_{ m1} a_{ m2} ... a_{ mn}]

Another denotation for matrix A is A = [a_{ ij}] for 1<= i <= m and 1<= j<= n.

The **order** of a matrix having m rows and n columns is m x n.

If the number of rows of a matrix is equal to its number of columns, it is a **square** matrix and its **main diagonal** entries are : a_{ 11}, a_{ 22}, a_{ 33} . . .

Only matrices of the same order (same number of rows and same number of columns) may be added by adding corresponding elements.

Example:

1.
[2 -3] [7 3] [2 + 7 -3 + 3] [9 0]

[4 5] + [-2 7] = [4 + (-2) 5 + 7] = [2 12]

[6 0] [1 -5] [6 + 1 0 + (-5)] [7 -5]

2.
[2 -3] [7 3 5]

[4 5] + [-2 7 -9]

[6 0] [1 -5 0]

The addition above is undefined since the two matrices do not have the same number of columns.

To multiply matrix A of order m x n by a scalar k, we multiply each element of matrix A by k to obtain another matrix of the same order.

Example:

1.
[2 -3 7]

A = [4 5 -3]

[6 0 -1]

Matrix - 2 A is given by
[-2*2 -2*-3 -2*7] [-4 6 -14]

- 2 A = [-2*4 -2*5 -2*-3] = [-8 -10 6]

[-2*6 -2*0 -2*-1] [-12 0 2]

If A = [a_{ ij}] is a matrix of order m x n and B = [b_{ ij}] a matrix of order n x p, then the product C = AB of the two matrices is a matrix of order m x p matrix defined.

C = [c_{ ij}] where c_{ ij} is given by

c_{ ij} = a_{ i1}b_{ 1j} + a_{ i2}b_{ 2j} + a_{ i3}b_{ 3j} + . . . + a_{ in}b_{ nj}

**Note that a multiplication of two matrices AB is defined only if the number of columns of matrix A is equal to the number of rows of matrix B.**

Example: Find the product AB where A and B are matrices given by

1.
[2 -3] [1 0]

A = [4 5] , B = [-2 1]

[6 0]

Solution

The product AB is defined since A is a 3 x 2 matrix and has __2 columns__ and B is a 2 x 2 matrix and has __2 rows__. To find elements of the product C = AB, multiply each row of A by each column of B.

[2 -3][1 0]

C = AB = [4 5][-2 1]

[6 0]

[(2)(1) + (-3)(-2) (2)(0)+(-3)(1)] [8 -3]

= [(4)(1) + (5)(-2) (4)(0)+(5)(1)] = [-6 5]

[(6)(1) + (0)(-2) (6)(0)+(0)(1)] [6 0]

You may want to explore more the multiplication of matrices using an applet.

__Exercises__

1 . What are the number of rows, the number of columns and the order of each matrix below? Which of these matrices is a square matrix?

a) [2 3] b) [-2 0 5] c) [0 0 4]

[-2 5] [4 -2 6]

[0 9] [-1 2 9]

2 . Find matrices

a) A + B

b) C^{ 2}

c) A + BC

d) A + CB

if possible. Matrices A, B and C are given by

[2 3] [-2 4] [2 1]

A = [-2 5] B = [3 1] C = [0 -1]

[0 9] [0 -8]

__Answers__

1.

a) Matrix below has 3 rows and 2 colums hence its order is 3 x 2. Since the number of rows and the number of columns are not equal the matrix is not square.

[2 3]

[-2 5]

[0 9]

b) Matrix below has 1 row and 3 columns, its order is 1 x 3. It is not a square matrix since the number of rows and the number of columns are different.
[-2 0 5]

c) Matrix below has 3 rows and 3 columns, its order is 3 x 3 and it is a square matrix since the number of rows and the number of columns are equal.
[0 0 4]

[4 -2 6]

[-1 2 9]

2.
[0 7]

A + B = [1 6]

[0 1]

b)
C^{ 2}= C C = [2 1] [2 1] = [4 1]

[0 -1] [0 -1] [0 1]

c)
[2 3] [-2 4][2 1]

A + BC = [-2 5] + [3 1][0 -1]

[0 9] [0 -8]

We calculate the product BC first and then add.
[2 3] [-4 -6] [-2 -3]

A + BC = [-2 5] + [6 2] = [4 7]

[0 9] [0 8] [0 17]

d)
[2 3] [2 1][-2 4]

A + CB = [-2 5] + [0 -1][3 1] = undefined

[0 9] [0 -8]

The product CB is undefined since the number of columns of C is 2 and the number of rows of B is 3. They are not equal.
__Exercises__

1. What is the order of each matrix.

A = [1 ]

B = [1 2 4]

[2 -1 0]

2. Add matrices A and B given below.

B = [1 0 4 -8]

[2 -1 4 3]

A = [-1 2 4 -8]

[4 -1 2 2]

3. Multiply matrices A and B given below.

[0 0 4] [1 -1 2]

A = [4 -2 6] B = [0 2 -1]

[-1 2 9] [2 2 3]

__Answers__

1. order of matrix A is 1 X 1 , the order of matrix B is 2 X 3

2.

A + B = [0 2 8 -16]

[6 -2 6 5]

3.
[8 8 12]

A . B = [16 4 28]

[17 23 23]

More pages and references related to matrices.
multiplication of matrices using an applet.

Find Inverse Matrix - Calculator.

Find Inverse of 3 by 3 Matrix - Calculator.