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Matrices and Determinants

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Table of Contents

1. Introduction
2. Division of Matrices
3. Determinants
4. Example
5. Summary
6. Frequently Asked Questions (FAQs)

September 8, 2020,     

Reading Time: 7 mins       

Introduction

This article explains about complex operations that can be performed on matrices, their properties, and Matrix’s extensive utility in various real-time applications used across the world.
It also consists of determinants and determinants properties. 
For basics of Matrices, please refer the article titled “Basics of Matrices


We have covered the addition, subtraction and multiplication operations in Basics of Matrices.
Now, we will see how division is performed on matrices.


Division of Matrices

We can divide one matrix by another matrix. But there is a catch!
We perform division operation on matrices indirectly. We do not do a direct division.
For instance, If we have to divide Matrix A by Matrix B, we do not do A/B. 
We multiply A to the inverse of B.
A × (1/B) 
1/B is the inverse of Matrix B.
Inverse of a matrix is nothing but the reciprocal of the Matrix.

Which brings us to the question – what is a reciprocal? 
Let us see an example - 
Reciprocal of the number 2 is 1 / 2. It can also be written as 2 to the power of (-1)
Which is denoted as 2-1.
So, when we say a matrix is inversed, we first find the reciprocal of the matrix, or in other words – Matrix to the power of (-1) which is denoted as M-1.
But before we proceed to see how to find the inverse of a Matrix, we also need to learn about Determinants.


Determinants

A determinant is a number that is calculated from a square Matrix.
The determinant of a square matrix is denoted by two vertical lines on either side, like the one shown below.

Determinant

Determinant of 2 x 2 Matrix

The Determinant of a 2 x 2 matrix is calculated as shown below.

matrix

Multiply the elements a11 and a22. Similarly multiply the elements a12 and a21.
a11 = 6 ; a22 = 8 ; a12 = 4 ; a21 = 10
So a11 × a22 = 6 × 8 = 48
and a12 × a21 = 4 × 10 = 40
Now subtract the second product from the first product.
48 – 40 = 8
So, the determinant for the matrix A is 8.
This is the method to calculate the determinant of a 2x2 matrix.

 

Determinant of 3 x 3 Matrix

The Determinant of a 3 x 3 matrix is calculated as shown in the following steps.

  1. Multiply an element by the determinant of the 2 x 2 matrix that is not in the concerned element’s row or column
  2. Repeat the process for all the elements present in the same row or column as the first element. Add a minus sign to the determinant value for the second element in the row (or column)
  3. Add all the values

An example is shown here,

matrix

Let us do the steps as mentioned above.
Let us consider the elements in the first row: 4,3, 0, and find the determinants corresponding to submatrices of these elements.

  1. 4 × [(5 × 2) - (6 × 0)]  
  2. -3 × [(1 × 2) - (6  × 6)]
  3. 0 × [(1 × 0) - (5 × 6)]

Adding all the above values, we get the determinant as 40 – 102 + 0 = 62
Please remember that for a determinant of a 3 x 3 matrix, elements of any row/column can be chosen and the above-mentioned steps followed in a similar fashion.

 

Determinant of a 4 x 4 Matrix or Higher

The same method is used in the determinant of a 4 X 4 matrix or matrices of higher order.

  1. Choose a row / column.
  2. Find the determinants of all the submatrices excluding the particular row and column, assigning alternate positive and negative signs.

Please also remember that the first element (1,1) is always positive.
Adding all these determinant values gives us the determinant of the 4 x 4 matrix.

Now let us go back to finding the Inverse of a matrix. 

 

Inverse of a Matrix

An example is shown below.                     

inverse of a matrix

This is an example of a 2 x 2 matrix. To get the inverse, we interchange the numbers a and c and reverse the sign of the other two numbers. So, b and d become –b and –d. Once this is done, the answer is multiplied by 1/Determinant of the matrix in question.

matrix

Now coming back to dividing one matrix A by another matrix B, 

  1. First, find the inverse of the Matrix B (following the same method as explained above)
  2. Then, multiply it by the first Matrix A (following the same method as explained in the Multiplication section).

A × (1/B)


Example

Q. Why do we need Matrix? How are they used in our day to day lives?
Let us see a practical example of how Matrix can be used to find answers to a mathematical problem in a simple way.
Here is an example problem statement:
Rahul goes to a shop and pays Re 34 for 2 erasers, 1 pencil, and 4 sharpeners. His friend Sam pays Re 35 for 3 erasers, 2 pencils and 2 sharpeners. Nancy pays Re 49 for 5 erasers, 3 pencils and 2 sharpeners. What is the price of each eraser, sharpener, and pencil?

Let us denote the items in the following method
Eraser – e
Pencil – p
Sharpener – s
As per the question we can say,
2e + 1p + 4s = 34
3e + 2p + 2s = 35
5e + 3p + 2s = 49
These linear equations can be represented easily in a matrix like this -
       
To find the value of e, p, and s, we need to divide the cost Matrix by the quantity Matrix-like shown below.

                         
When we solve this following all the steps explained in this article, we get the answer for e, p and s as
 e = 3
 p = 8
 s = 5


Summary

To conclude, Matrix is an important concept used in a wide range of fields. As seen in this article, Matrix is used in many high-profile apps including graphic imagery. At the same time, the concepts of Matrix operations are quite easy to understand.

 

Written by Gayathri Sivasubramanian, Cuemath Teacher


Frequently Asked Questions (FAQs)

What are the uses/applications of Matrices?

Matrix is widely used in many areas of work in Mathematics as well as other Sciences:

  • Seismic Surveys
  • Classical Mechanics
  • Quantum Mechanics
  • Aerodynamic Computations
  • Statistical Analysis
  • Robotics and automation
  • graphics software

and in so many more.

What is the Transpose of a Matrix?

We get the Transpose of a Matrix when we interchange the rows and columns.
There is an example shown below:

matrix

The Transpose of Matrix B is,

Transpose of a matrix

From the definition of Transpose, I think it will make sense to conclude that, if we take a Transpose of an already transposed Matrix, we get the original Matrix. 

What is an Orthogonal Matrix?

A matrix A is called Orthogonal if the product of the matrix and its transpose is equal to the Identity matrix.
i.e.  A x AT = I
In other words, the inverse of the matrix is its Transpose
i.e. AT = A–1
An orthogonal matrix is always a symmetric matrix.
Orthogonal matrices are widely used in vector related applications.
Given below is an example of an Orthogonal Matrix.

matrix

What is an Adjoint of a Matrix?

The Adjoint of a Matrix is obtained by transposing the Matrix of its cofactors.
What is a Cofactor Matrix?
Let us again see an example here. Consider the matrix A shown below.

matrix  

The cofactor matrix is calculated by finding the determinant of the values which exclude the row and column in which the concerned element is present.
For example, the cofactor of the element 2 in the position (1,1) is the determinant 

Deteminant

Which is equal to 4 – 6 = -2
Later, an alternate negative or positive sign is put in front of the determinant value, depending upon the position of the element. The element in (1,1) is given a positive sign. Element (1,2) is given a negative sign and element (1,3) is again given a positive sign and so on.
By this method, the cofactor matrix of the Matrix A is calculated as

matrix

The transpose of this matrix gives the Adjoint of the original Matrix, as shown below.

matrix

What is a Singular Matrix?

A matrix is called Singular if its determinant is zero. This also means that a singular matrix does not have an inverse.
Let us see an example here,

The determinant of this matrix is 6 - 6 = 0. Hence this is a Singular Matrix.

matrix

 

What is a Rotation Matrix?

A rotation matrix is used to represent rotation in the counter-clockwise direction about the x-y plane. 
The matrix given below represents a counter-clockwise rotation by an angle of α radians.

Rotation matrix

What is an Identity Matrix?

A square matrix with all 1s in the diagonal and the rest of the elements as 0 is called an Identity Matrix. An identity matrix looks like the one shown below.

Identity matrix

What is the Rank of a Matrix?

The rank of a matrix denotes the number of unique rows/columns in a matrix. This is determined by the following method. 
For a 3 x 3 matrix, 

  • The rank is 1: If the determinant of the matrix is not zero (Non- singular)
  • The rank is 2: If the matrix is singular, but the at least one of its submatrices has a non-zero determinant
  • The rank is 1: If every submatrix is a singular matrix (determinant is zero)

A matrix is said to be Symmetric if it is equal to its Transpose.
A = AT

What is a Diagonal Matrix?

A matrix is said to be Diagonal if all the elements except the diagonal are zero.

Diagonal matrix


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