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Hi all,
Today we are going to learn one of the main concepts of Matrices (Mathematics) for GATE Exam. The concept is how to find the nature of solution for the given set of equations using Matrices. If you already know the complete concept then you can skip this article. This concept is important for gate exam. We can expect one question for GATE Exam.
In the GATE exam we can expect below kind of questions.
(Q1) Consider the system of simultaneous equations
x+2y+z=6
2x+y+2z=6
x+y+z=5
This system has
(A)Unique solution (B) Infinite number of solutions (C) No solution (D) Exactly two solutions

(Q2) For what values of a,b the following simultaneous equations have an infinite number of solutions
x+y+z=5
x+3y+3z=9
x+2y+az=b
(A)a=2,b=7 (B)a=3,b=8 (C) a=8,b=3 (D)a=7,b=2

We are going to learn this concept by using step by step process.
First we have to understand the concept behind this and then we will try to solve the problems. I believe if we understand the inner meaning then we are going to remember it forever.

No Solution Scenario:

For example, one shopkeeper gave us a challenge.
The price of one Apple and one Mango is Rs.20.
The price of two Apples and Two Mangos is Rs.50.
We need to find the price of Apple & Mango individually.
Let us say A for Apple, M for Mango.
According to shopkeeper
1A+1M=20 -->(1)
2A+2M=50 -->(2)
Take 2 common out on left side of equality sign in equation (2)
Then,
2(1A+1M)=50
-->1A+1M=50/2=25
After reducing second equation to short form our equations are
1A+1M=20 -->(1)
1A+1M=25 -->(2)
Do you think we can find the prices from the above two equations? No. We can’t. The sum of prices of one Apple and one Mango should always be same.
But in the first equation he is saying that price of one Apple and one Mango is Rs.20.
In the second equation he is saying that price of one Apple and one Mango is Rs.25 indirectly.
This is not possible (Don’t say one is Bigger Apple and other is Smaller Apple).
The above scenario will not have any solution. This is No solution scenario.

Infinite number of solutions scenario:

Let us take the same kind of example like previous example.
The price of one Apple and one Mango is Rs.20.
The price of two Apples and Two Mangos is Rs.40.
We need to find the price of Apple & Mango individually.
Say A for Apple, M for Mango.
According to shopkeeper
1A+1M=20 -->(1)
2A+2M=40 -->(2)
Take 2 common out on left side of equality sign in equation (2)
Then,
2(1A+1M)=40
-->1A+1M=40/2=20
After reducing 2nd equation to short form our equations are
1A+1M=20 -->(1)
1A+1M=20 -->(2)
In the beginning it looked like we have two equations to find the values for two variables. But after reducing one equation both of them become same. That means the given both equations are same. So consider any one equation.
Let us consider 1st equation
1A+1M=20
If we assume price of Mango=Rs.10 then price of Apple will become Rs.10.
If we assume price of Mango=Rs.5 then price of Apple will become Rs.15.
If we assume price of Mango=Rs.7 then price of Apple will become Rs.13.
Like this we will be having infinite combinations to satisfy the given equation.
This scenario is Infinite number of solutions.

Unique Solution Scenario:

Again consider same kind of example.
The price of one Apple and one Mango is Rs.20.
The price of two Apples and 3 Mangos is Rs.50.
We need to find the price of Apple & Mango individually.
Say A for Apple, M for Mango.
1A+1M=20 -->(1)
2A+3M=50 -->(2)
If we solve the above two equations we will get A=10, M=10.
Other than A=10, M=10 will not satisfy the given equations. At last we calculated the prices of Apple & Mango individually.
This is Unique Solution scenario.

Now we are clear with the concept behind this. Solving directly the given equations will not be easy. But if you practice more problems you can directly say the scenario.
To make our life easier we use Matrices to find these scenarios.
We have to use three parameters to decide the scenario.
(1) Rank(A)
(2) Rank(A|B)
(3) Number of variables
Let us take the example given at the starting of this article.
(Q1) Consider the system of simultaneous equations
x+2y+z=6
2x+y+2z=6
x+y+z=5
This system has
(A)Unique solution (B) Infinite number of solutions (C) No solution (D) Exactly two solutions

Solution:
First we have to write the given equations in Matrix form like below.
Matrices-Solution-Nature-1
We can interchange the order of equations. Those are just equations. We can take any equation from the given set of equations as first or second or third equations. That is completely our choice. Normally we change the order if we get first element of first row of A matrix is zero.

Now we need to find the three parameters.

(1) Rank(A|B)
(2) Rank(A)
(3) Number of variables

We can find Rank(A) from Rank(A|B). So first we find Rank(A|B)
A|B means we need to place Matrices A and B side by side like below image.
Matrices-Solution-Nature-1
Now to find Rank(A|B) we need to use row echelon form.
Row echelon form means all the elements below the diagonal of a matrix should be zero. (Elements below the first element of first row, elements below the second element of second row and so on should be zero in row echelon form.)
Matrices-Solution-Nature-3
We have to use some row transformations to achieve row echelon form.
First we have to make the elements below the first element of first row. See below images to how we are doing that. We have to apply the transformation on complete row.


Matrices-Solution-Nature-4


Matrices-Solution-Nature-5
Now we made all the elements below the first element of first row as 0.
Now we have to make the elements below the second element of second row as 0. See below image.
Matrices-Solution-Nature-6
We do not have any elements below the third element of third row. We are done with row echelon form.
Our final row echelon form of A|B matrix is
Matrices-Solution-Nature-7

Now we have to find Rank(A), Rank(A|B), Number of variables
Rank means number of non zero rows after performing row echelon form.
Rank(A)
Consider matrix A only from the above image, we have at least one non zero element only in 2 rows in Matrix A of Matrix A|B.
So, Rank(A)=2
Rank(A|B):
Consider whole matrix A|B from the above image, we have at least one non zero element in 3 rows in Matrix A|B.
So, Rank(A|B)=3
Number of variables:
We have 3 variables.(From Matrix X)

Solution Nature scenarios using Matrices:
Unique Solution(Consistent):
Rank(A)=Rank(A|B)=Number of variables
Infinite Number of Solutions(Consistent):
Rank(A)=Rank(A|B)<Number of variables
No Solution(Inconsistent):
Rank(A)≠Rank(A|B)

For our problem
Rank(A)≠Rank(A|B)
So we do not have any solution existed for our equations given in the problem. (No Solution)
Answer is C.

For the all these kind of problems we need to follow the above said procedure. If we practice more problems we can directly say solution nature.

See also:

We will solve the another example problem given at the top of this article.

(Q2) For what values of a,b the following simultaneous equations have an infinite number of solutions
x+y+z=5
x+3y+3z=9
x+2y+az=b
(A)a=2,b=7 (B)a=3,b=8 (C) a=8,b=3 (D)a=7,b=2

Solution:
We will be following the same process we have followed in previous problem.
First write in matrix form.
Matrices-Solution-Nature-8
Now we need to find Rank(A|B), Rank(A), Number of variables.
For that first A|B matrix then apply row echelon form transformations.
Matrices-Solution-Nature-9
Matrices-Solution-Nature-10
Our final A|B matrix after applying row echelon form transfermations is
Matrices-Solution-Nature-11
In the question it is given that the infinite number of solutions exist for the given set of equations.
That means we have the below case.
Rank(A)=Rank(A|B)<Number of variables.
Number of variables=3 (From Matrix X)
That means Rank(A) & Rank(A|B) should be less than 3 and also equal.
We are already having 2 non zero rows in reduced A and reduced A|B matrices.
If third row also become non zero then the above condition will not satisfy. So third row has to become zero.
So,
a-2=0 and b-7=0
-->a=2 and b=7
Then Rank(A)=2, Rank(A|B)=2
Infinite number of solutions condition is satisfied.
When a=2, b=7 then only we will be having infinite number of solutions for the given set of equations.

Answer is A.

Try to solve the more problems. You will get practiced.
If you have any doubt please comment below.

Thank you.
All the best.
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