111.Aromaticity (4) – Matrix

Before proceeding further, it is imperative that we understand some basic mathematical concepts required to study this theory. We are not going to elaborately study the mathematics – just enough to comprehend the crux of HMO theory.

Matrix

Matrix is a mathematical concept which is used in HMO theory. A matrix contains rows and columns of numbers , inside a square bracket.
e.g.-  Shown in the figure below is a matrix A.
This matrix has three rows and three columns and thus the order of this matrix is 3 × 3.

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The above matrix is also represented as Aij as shown below- 

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Thus, the above matrix is a A33 matrix.This means that the matrix has three rows and three columns.

A matrix that has same no of rows and columns is called a square matrix. Thus, the above  A33 is a square matrix.

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Each number in the matrix is called an ‘element of the matrix’. Any number in the matrix can be denoted as aij shown below –
Matrix concept helps us solve a number of equations with different variables faster.( For more on what variables are refer to post no 5). Generally, we need to take a single equation and work upon it to get the solution. However, with matrix mathematics we can solve a set of equations all at once.

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Polynomials

The equations are called polynomials. Polynomials are algebraic equations involving numbers and variables.

e.g.- x2 + 4x +7 =0 is a polynomial equation which has x as a variable.

The highest exponent of the equation is called the degree of the polynomial and it tells us how many roots that equation has. Thus, if the polynomial has a degree two , then it means that it has two roots.

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Identity matrix

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Understanding this matrix is very important to our study of HMO theory. An identity matrix is a square matrix, which has ones (1) on its diagonals and all zeros(0) elsewhere.
Square matrix means a matrix having equal no of rows and columns (n×n). As seen in the above figure, the identity matrix has 4 rows and 4 columns and thus it is a square matrix.


 

Transpose of a matrix.

In algebra, transpose is an operator which flips a matrix over its diagonal. Thus, after transposing, the rows become columns and vice versa.

e.g.– If A is a matrix as shown below, then Ais its transpose matrix

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Orthonormal Matrix

An orthonormal matrix is a square matrix , where,

AT. A = A. AT = I , I = Identity matrix.


Determinant

Determinant of s matrix tells us a lot about that matrix and it helps in calculations. The determinant of matrix A is shown by  |A| . It is imperative that the matrix whose determinant has to be calculated is a square matrix.

A]Calculating determinant of a 2×2 matrix – 

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Consider a 2×2 matrix as shown in the following figure –
The determinant of this matrix is given by |A| = ad − bc.

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e.g.- 
The determinant will be,

 
|B|= 4×8 − 6×3
 = 32−18
 = 14

B]Calculating determinant of a 3×3 matrix – 

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Consider the following matrix –
The determinant of this matrix can be calculated as follows –

  • Multiply a by the determinant of the 2×2 matrix that is not in a‘s row or column(e,f,j,k part ).
  • repeat the same for b, and for c
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Add them, but remember the minus in front of the b.

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e.g.- The determinant of the above matrix is –

|A|= 6×(−2×7 − 5×8) − 1×(4×7 − 5×2) + 1×(4×8 − (−2×2))
= 6×(−54) − 1×(18) + 1×(36)
= −306.


Matrices help in producing graphic images / graphs from linear transformations. All computers have embedded matrix mathematics in them which is used to produce graphic forms from data. It is always easy to understand graphic data and so this concept is very useful.

In the next chapter we will see how matrices are used with the Huckel theory. Till then,

Be a perpetual student of life and keep learning…

Good day !

 

References and further reading –

1.https://en.wikipedia.org/wiki/Transpose

2.https://www.mathsisfun.com/algebra/matrix-determinant.html

 

 

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