Matrices

The concept and theory of matrices in mathematics was first coined and proposed by Arthur Cayley. Matrices are useful in expressing numerical information in a very compact form. They are used effectively in expressing different operators. Due to this, they form an essential part of Economics, statistics and computer science.

Definition

A rectangular arrangement of mn numbers in m rows and n columns, surrounded in [ ] or ( ) is called a matrix of order m by n.

Order of matrix is written as m*n ,which is read as m by n.

A matrix by itself has no value of its own, each member of the matrix is called an element of the matrix. 

Matrices are denoted by the capital alphabets such as A , B , C and so on while the elements in them are denoted by a_ij , b_ij and so on…

Now this matrix A has 3 rows and 3 columns. Therefore, we can say that the order of this particular matrix is 3 x 3 which as read as three by three. 

Also, the values that we can see inside the brackets are nothing but elements of this particular matrix as mentioned above. The element a11 is 2, which is nothing but elements in row 1 and column 1. Similarly, the element value in a32 = -2 and so on we can write down the value of every element.

To express the matrix in general form let us consider the matrix A 

 A =a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ = 2 -3 9 1 0 -7 4 -2 1  

Readers must take note that aij  can go up to mn number.

Types of Matrix 

1. Row matrix: Any matrix having only one row is called a row matrix. 

Example: 2 -3 this is a 1 by 2 matrix; one can construct 1 by 3 matrix too.

2. Column matrix: Any matrix having only one column is called as column matrix.

Example:  -9 5 this is a 2 by 1 matrix; one can construct 3 by 1 matrix too. Here care must be taken that the matrix has only column and by no means more than that. 

3. Null or zero matrix: The peculiar matrix where every element is zero is known as null or zero matrix. In such matrix one has the liberty to construct any matrix with any number of rows and columns, only care must be taken is that the elements in that matrix must be zero only.

Example: 0 0 0 0

4. Square matrix: Any matrix having equal number of rows and columns is called a square matrix.

Example: 2 -3 9 1 0 -7 4 -2 1 3 by 3 matrix.

5. Diagonal matrix: Any square matrix in which every non-diagonal element is zero is called a diagonal matrix.

Consider the below matrix to understand diagonal first, 

  2 -3 9 -1 0 -7 4 -2 1 here, the elemental values 2, 0, 1 are known as the diagonal element. 

So as per our definition stated above, lets construct another 3 by 3 matrices as follows:

A = 1 0 0 0 2 0 0 0 3 here the diagonal is 1,2,3.

6. Scalar matrix:  A diagonal matrix having all the diagonal elements same is known as scalar matrix.

Example: 2 -3 9 1 2 -7 4 -2 2  

7. Unit or identity matrix: A scalar matrix having diagonal elements as unity is known as unit matrix or an identity matrix.

Example:   1 0 0 0 1 0 0 0 1

8. Upper triangular matrix: A square matrix in which every element below the diagonal is zero is known as upper triangular matrix.

Example: A = 4 -2 5 0 0 3 0 0 2

Readers must take note that, matrix A = [ aij] n*n is upper triangular if aij = 0 for all i > j.

9. Lower triangular matrix: It’s exactly opposite as compared to upper triangular matrix. Here every element above the diagonal is zero.

 Example: 4 0 0 -1 9 0 7 2 5

10 Triangular matrix: Any square matrix is said to be triangular matrix if it is an upper triangular or a lower triangular matrix.

11. Symmetric matrix: A square matrix A = [ a ij]n*n in which aij = aji for all i and j , is called a symmetric matrix.

Example: B = -3 1 1 8

12. Skew-Symmetric Matrix:  A square matrix A = [ a ij]n*n in which aij = – aji for all i and j , is called a skew symmetric matrix. Here each diagonal element is zero.

Example:  A = 0 5 -5 0

13. Determinant of a matrix: let’s say A is a square matrix then the same arrangement of the elements of A also gives us a determinant, by replacing square brackets by vertical bars. It is denoted by │ A │ or det(A).

If A = 2 6 -5 4

Determinant of A = │A│ = 2 6 -5 4

And the determinant is solved as,  4*2 – 6*(-5) = 8 +30 = 38

If │A│ = 0 then it becomes a singular matrix

Properties of Determinants:

• The value of determinant remains unchanged if its rows are turned into columns and columns are turned into columns are turned into rows.
• If any two rows or columns of the determinant are interchanged then the value of determinant changes its sign.
• If any two rows or columns of a determinant are same or identical then the value of determinant is always zero.
• If each element of a row or column of determinant is multiplied by a constant k then the value of the new determinant is k times the value of given determinant. 
• If each element of a row or column is expressed as the sum of two numbers then the determinant can be expressed as sum of two determinants.

Transpose of a matrix: 

The matrix obtained by interchanging rows and columns of matrix A is called Transpose of matrix A, and is denoted by AT or A’.

Example: A = -1 5 3 -2 4 7

Then AT = -1 3 4 5 -2 7

Conclusion

Matrices are useful in expressing numerical information in compact form. They are effectively used in expressing different operators. Due to this, they form an essential part of Economics, statistics and computer science. They are used in multiple ways to solve system of equations. Matrices are also used by physicist , mathematicians to organise data and study complex phenomenons.