Matrix Addition and Multiplication

A tutorial on the addition and multiplication of matrices.

Definition of a Matrix

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 . . .



Addition of two Matrices

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.

Scalar Multiplication of a Matrix

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]

Multiplication of two Matrices.

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 i1b 1j + a i2b 2j + a i3b 3j + . . . + a inb 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.