Last Updated on: 15th January 2024, 11:45 am

Matrices

Matrices – Meaning

Most commonly, a matrix over a field F is a rectangular array of scalars, each of which is a member of F (we will discuss about Matrix of Real Numbers only).

The numbers in the Matrix are called member, and is represented by a_i,j, where i & j are the row and column (i^th row & j^th column) position of the member in the array.

Order of a Matrix

A matrix having m rows and n columns is called a matrix of order (or Dimension) m × n or simply m × n matrix (read as an m by n matrix). So, a matrix of order has m × n elements. For example,

A matrix of order 3 × 2 matrix has 3 × 2=6. A matrix of order 1 × 1 has only one element. A matrix of order 1 × 3 has 1 row and 3 columns consisting of 3 elements. only one element. Similarly, A matrix of order 3 × 1 has 3 rows and 1 column consisting of 3 elements.

Format of a Matrix

In general, an m × n matrix has the following rectangular array:|
a₁₁      a₁₂      a₁₃  ….. a_1n
a₂₁      a₂₂      a₂₃  ….. a_2n
….      …..    …..  ….. …..
a_m1      a_m2      a_m3  ….. a_mn

So, the i^th row has elements a_i1, a_i2, a_i3,…, a_in, while the jth column has elements a_1j, a_2j, a_3j,…a_mj ,

In general, a_ij, is an element lying in the i^th row and j^th column, referred as (i, j)^th element of the matrix. The number of elements in an m × n matrix will be equal to m×n.

Let us see, how we can represent the following table of Men and Women in 3 Clubs in Matrix Format :
Club 1   10 men     6 women
Club 2   8 men       5 women
Club 3   11 men     8 women

There are 3 clubs and 2 types of people (men, women). So, the numbers can be represented in a Matrix of 3×2
 10  6
I 8   5 I
 11  8

Types of Matrices

Column Matrix

A Matrix having one column only, like: i.e, A = [a_ij]_{m × 1} is a column matrix of order m × 1.

Row Matrix
A Matrix having one Row only, like: .i.e, B = [bij]_{1 × n}is a row matrix of order 1 × n.  

Square Matrix
A Matrix in which the number of rows and columns are equal is called a square matrix. Thus, an m × n matrix is said to be a square matrix if m = n.  i.e., A = [aij]_m×mis a square matrix of order m.

So,    -1/2    8/3
  I -7          12 I
   -9    11     16
is a square matrix of order 3

Diagonals of Square Matrix
The elements a₁₁, a₂₂, … a_nn constitute the Diagonal of Matrix. So, in the above Matrix, the elements Diagonal of the Matrix are , ,  16. All other elements of the Matrix are called non-diagonal elements.

Diagonal matrix
A square matrix B = [b_ij] _{m × m}is said to be a diagonal matrix if all its non-diagonal elements are zero, that is a matrix B = [b_ij] _{m × m} is said to be a diagonal matrix if b_ij = 0, when i ≠ j.

So, the matrix A=[5] is Diagonal matrix of order 1, B= I  2 0 I
                        I 0  I
is a Diagonal matrix of order 2,

C=  1  0  0
  I 0  4  0 I is a Diagonal matrix of order 3.
   0  0 -7   

Scalar Matrix
A Diagonal Matrix is said to be a scalar matrix if its diagonal elements are equal. So, a square matrix B = [bij] _n×nis said to be a scalar matrix if bij = 0, when I ≠ j. So, bij = k, when i = j, for some constant k.
So, the matrix A= [5] is Scalar matrix of order 1, B= I 2  0 I
                       I 0  2 I
is a Square matrix of order 2,

C= 4  0  0
  I 0  4  0   I is a Scalar matrix of order 3.
  0  0  4

Identity Matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other words, the square matrix A = [a_ij] _{n × n}is an identity matrix, if 1 if a_ij = 1 (when i=j) and a_ij = 0 (when i ≠ j).

So, A= [1] is Identity matrix of order 1, B= I 1 0 I  is Identity matrix of order 2, C=  1  0  0  
                  I 0 1 I               I 0  1  0 I
                                    0  0  1
is Identity matrix of order 3. So, Identity matrix is a scalar matrix when k = 1. So, every identity matrix is a scalar matrix (but every scalar matrix is not an Identity matrix).

Zero Matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero. For example, A=[0] is zero matrix of order 1, B=I 0 0 I is zero matrix of order 2, C=1 0 0 is zero matrix of order 3. 
            I 0 0 I            I 0 1 0 I
                           0 0  1

Equality of matrices
Two matrices A = [a_ij] and B = [b_ij] are said to be equal if (i) they are of the same order and (ii) each element of A is equal to the corresponding element of B, that is a_ij = b_ij for all i and j.

So, Matrix A= I 1  2 I and Matrix B=I 1 2 I are equal as all the corresponding elements are
      I 3 4  I       I 3 4 I 
equal. But Matrix A= I 3 4 I and Matrix B= I 1 2 I are not equal. If Matrix A and Matrix B are
           I 1 2 I       I 3 4 I
equal, we may write A=B

Negative of Matrix
Negative of a Matrix is obtained by replacing each element a_ij with – (a_ij)_. So negative of matrix
A= I   I
  I -3  4 I   
may be written as  B= I   I
          I 3   -4 I 
Here B=-A, or A=-B.

Addition of Matrices
Two matrices have to be of the same order. So, Matrix A= I 1 2 3 I and Matrix B= I 7 8 9 I 
                          I 4 5 6 I        I 10 11 12 I
be can added but a Matrix A= I 1 2 3 I (of order 3×2) and a Matrix B= I 9 10 I (of order 2×3)
             I 4 5 6 I               I 11 12 I
cannot be added. In addition of Matrix, the matrix is created by adding the corresponding elements of both the Matrix. So, by adding 2 matrices

A= I 1 2 3 I and B= I 7 8 9 I , we get Matrix C= I 1+7   2+8  3+9 I,
  I 4 5 6 I    I10 11 12 I          I 4+10  5+11  6+12 I

Or, C= I 8  10 12 I
    I14 16 18 I

Difference of Matrices

If A = [a_ij], B = [b_ij] are two matrices of the same order, say m × n, then difference A – B is defined as a matrix D = [d_ij], where d_ij = a_ij – b_ij, for all value of i and j. In other words, D = A – B = A + (–1) B, that is sum of the matrix
A and the matrix – B (i.e negative of Matrix B).
For Matrix A= I 7 8 9 I, and B= I 1  2  4 I we get Matrix
       I -10 11 12 I    I -4 5  -6 I

C= I 7-1   8-2  9-4  I  , or C= I 6 6 5 I
  I -10-(-4)  11-5  12-(-6) I     I -6 6 18 I