Introduction To Determinants

Determinants are very important concept related to square matrix, and usually it is simple to calculate one while dealing with the system of linear equations. Our attempt here is to understand determinants properly so that all related concepts becomes easy and unforgettable. So we begin with a gentle introduction to determinants of matrices.

What are Determinants ?

Imagine there are no matrices and you must solve the system of linear equation with simple algebra. For example,

Consider the following system of linear equation with just one equation.

a_{11}x_1 = b_1

To solve this equation using algebra method divide both sides by constant a_{11}

x_1 = \frac{\strut b_1}{\strut a_{11}}

The element a_{11} is the determinant in the equation as a denominator. Provided a_{11} \neq 0 

Otherwise, the system has no solution.

A determinant is zero does not always means that the system of linear equation has no solution, sometimes it has infinite solutions. But, if determinant is 0, then the matrix is not invertible.

Let us take another example, where there are two linear system of equations with two unknowns.

a_{11}x_1 + a_{12}x_2 = b_1

a_{21}x_1 + a_{22}x_2 = b_2

When we solve for x_1 the equations become

x_1 = (b_1 - a_{12}x_2)/a_{11}

x_1 = (b_2 - a_{22}x_2)/a_{21}

Comparing both equations we get 

x_1 = a_{21}b_1 - a_{12}a_{21} = a_{11}b_2 - a_{11}a_{22}

 = a_{21}b_1 - a_{11}b_2 = a_{12}a_{21} - a_{11}a_{22}

 = (a_{21}b_1 - a_{11}b_2)/(a_{12}a_{21} - a_{11}a_{22})

The denominator should be a_{21}a_{13} - a_{11}a_{22} \neq 0, else the equation will collapse.

Generalizing the Denominator

From the above result, there is a pattern to the denominator and if you solve for x_2 then you will find out that the denominator is not changing and it remain the same. Hence, the denominator is determinant of the equation given that it is not equal to zero.

For a system of two equations and two unknowns , the determinant is the following.

a_{12}a_{21} - a_{11}a_{22}

Observe that in any term of determinant, the i corresponds to sequence of numbers i = 1,2,3,...,n and all j in the term is a permutation of the sequence 1,2,3,...,n. 
Therefore, we could generalize determinant as 

= \sum \pm a_{1\alpha}a_{2\beta}...a_{nv}

where i = 1,2,3,...,n determines the number of variables a_{ij} in a term.

Problem Of Negative And Positive Terms

Though we were able to generalize any term of the determinant with n variables in each term. Some terms are positive and some are negative. The next question is to determine which are positive and which are negative terms where any single terms is given by the following.

= \sum \pm a_{1\alpha}a_{2\beta}...a_{nv}

Before that you need to be familiar with following concepts.

Inversion and Transposition

In the determinant term, i = 1,2,3, …,n in natural order and all j are a permutation of the natural order (1,2,3,…,n).

Given a permutation \alpha,\beta,\gamma,…,v if two indices are out of their natural order and greater index comes first and then lesser index, then it is called inversion.

Suppose that the natural order is $latex(12345)&s=1$ and the permutation is $latex(21345)&s=1$, here 2 and 1 is inversion as they are out of the natural order.

Suppose that the natural order is disturbed by swapping two numbers. If $latex(12543)&s=1$ is the permutation then only 3 and 5 is swapped. This is called transposition.

The number of inversion is unique and can be counted. If the inversions are even then permutation \alpha,\beta,\gamma,…,v belongs to even class, otherwise, odd class.

If a term belongs to odd class, then assign negative sign, else assign a positive sign if it belongs to even class.

Determinant of 3 x 3 Equations

Using the information above let us systematically find determinant of 3 \times 3 equations. Since, n = 3 there will be n! = 3! = 6 terms with each term having 3 variables that is, a_{ij}.

Write down each permutation and the term of determinant.

a_{11}a_{22}a_{33} //permutation of j is (1 2 3)
a_{11}a_{23}a_{32} //permutation of j is (1 3 2)
a_{12}a_{21}a_{33} //permutation of j is (2 1 3)
a_{12}a_{23}a_{31} //permutation of j is (2 3 1)
a_{13}a_{21}a_{32} //permutation of j is (3 1 2)
a_{13}a_{22}a_{31} //permutation of j is (3 2 1)

We get following 

$latex =
a_{11}a_{22}a_{33}+a_{11}a_{23}a_{32}+a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}a_{13}a_{21}a_{32}+a_{13}a_{22}a_{31}&s=1$

We must look at the permutation and change the sign of terms that belong to odd class meaning permutations that have odd number of inversions.

= a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}

This is a faster method of finding determinants, but you must be careful in checking the number of inversions. Next article, we shall discuss about finding determinants by cross multiplications and some interesting properties of determinants.

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