Hello everyone, welcome back to a new video.
In today's video, we are going to do the part ahead of the
matrix.
We have already made videos on operations and matrices.
I hope you all have watched them.
Today we are doing a step ahead of that.
That is, we will see how to find out the determinant of any
matrix.
Now, what is the determinant of any matrix?
First of all, know that.
There is no determinant.
What was your matrix?
It was in rows and columns.
Solving those rows and columns in some way, we get a scalar
quantity.
That is a constant number.
So what do we call that?
The determinant of that matrix.
Determinant will not come in the form of a matrix.
A scalar quantity will come.
Similarly, every matrix will have a unique determinant.
There is no other determinant of any matrix other than 1.
There is always a unique determinant.
Now, what type of matrix can you find out the determinant
of? You can only find out the determinant of a square
matrix.
What was a square matrix?
In which the number of rows and columns were equal.
That was our square matrix.
So, we have to learn how to find the determinant of a
square matrix.
The maximum number of values that you can find the
determinant of, that is up to 3 x 3 matrix.
3 x 3 matrix.
So, let us first look at the matrix below that.
If we want to find the determinant of a square matrix, then
the smallest square matrix is 1 x 1 matrix.
1 x 1 matrix.
What will be the number of elements in this?
The product of elements is 1 x 1.
Number of elements is 1.
Only one row and one column.
That is our matrix.
What is coming in it?
Only one element is coming.
Now, that element can be anything.
Any rational number, any complex number, anything.
So, whatever will be the value in the determinant of the
matrix.
If we want to find the determinant of it, then first of
all, how will we denote the determinant?
We will denote the determinant.
Either you do like this.
Absolute value of A.
Write the name of the matrix between two vertical lines.
And how can you write?
Take three words before the determinant.
And what should I write in the bracket?
Name of the matrix.
A is the matrix.
Now, what will be the determinant?
I told you that the determinant will be a scalar value
related to that matrix.
Now, there is only one value in it.
So, whatever value is there in it, that will be the
determinant.
Like, if it is minus 5, then the determinant will be minus
5.
Similarly, if the value is positive, then it will be the
same.
If it is 2, then the determinant will be 2.
If it is 0, then the determinant will be 0.
So, in this way, you have seen that the determinant value
can be negative or positive.
Like, it can be 2 or 0.
0, negative and positive.
When we have negative and positive, then there is no
problem.
It is a non-zero value.
Then, we will say that matrix, if we have the determinant,
not equal to 0.
And, if it is a finite number, then there is no problem
with positive and negative.
If it is not equal to 0, then that matrix will be called
non-singular matrix.
And, we will use this matrix a lot in the future.
Non-singular matrix.
And, if the determinant of any matrix is 0, then we will
call it singular matrix.
We have seen from this example that it can be negative,
positive and 0.
Let's move to the second case.
It is of 1 x 1 order.
Now, the second case would be of 2 x 2 order matrix.
Now, how is it made of 2 x 2 order?
Two rows and two columns.
What will be the total elements? 2 x 2, that is 4.
So, let's make a matrix.
In which, we have 4 elements.
Suppose, it is this.
Simply, 2, minus 2, 3, 4 and 5.
If you want to find the determinant of this, then how will
you find it?
Okay? You must have heard of cross multiply. Cross sign.
Make a cross like this.
We have to cross multiply.
So, how will we make a cross? First, we will make the part
of this entry.
We will solve it on the main diagonal.
That is, we will multiply minus 2 by 5.
Minus 2 into 5.
Then, we will multiply 4 by 3.
Now, we have a cross like this.
So, we will multiply 4 by 3.
And, we will put a minus sign in the middle.
When you solve this, you will get a number.
Scalar quantity.
That will be the determinant of this matrix.
So, what will be written?
Minus 2 into 5.
That is, minus 10.
Minus 4 into 3.
12.
That will be minus 22.
What will be the determinant of this matrix?
Minus 22.
Okay? So, if I say the general form. Suppose, you have any
number.
A, B, C, D.
So, what will be their determinant?
We will do A from D.
A into D.
Now, look.
The sign of the matrix was this.
You had big square brackets.
What will be the sign of the determinant?
The sign that we put outside A.
Outside the matrix.
Vertical lines.
What will you do with that matrix? Instead of big square
brackets, draw vertical lines. So, what will be the
determinant?
I am asking you indirectly.
So, what will be the determinant?
Cross multiply.
A into D.
Then B into C.
C, B.
And what will you do in the middle?
Minus sign.
So, AD minus BC.
And what will be CB?
Determinant of this 2 x 2 matrix.
So, it is very easy to find the determinant of 2 x 2.
Now, we will learn 3 x 3.
3 x 3 is also very easy.
But, it is a little tricky.
There was a direct answer in the first one.
But, we got the answer in the second one.
In that, we have to do 1, 2, 3 things.
So, we will see that.
So, let us see.
3 x 3 matrix.
In 3 x 3, you will have 3 rows.
And 3 columns.
So, I will give the name of the matrix.
P.
I will make any matrix.
Suppose, this is your matrix.
This is your matrix.
Matrix of order 3 x 3.
Now, how will you find the determinant of this?
To find the determinant, we will take any row or column.
Any row or column.
Along any column, we have to expand this matrix.
We have to expand this matrix.
It is your choice.
You can take any particular row.
You can take any particular column.
It depends on you.
Now, I will explain to you.
Let us take any row.
Let us take row 1.
I will teach you from row 1.
If we take row 1,
If we take row 1, how will we expand along it?
How will we expand along it?
What are we doing?
We are expanding along row 1.
Now, I will tell you some things.
Have you seen row 1?
How many elements are there in row 1? 1, 2, 3.
There are 3 elements.
Give some space to 3 elements.
For example,
First element is 2.
Next element is 1.
Third element is 4.
Have you written?
We have given some space to 3 elements.
First was 2, then 1, then 4.
What is the next thing you have to do?
You have to put signs in front of them.
What are the signs?
Sign means, either you will put positive or negative sign.
When you will put positive sign and when you will put
negative sign, let's know that.
If you put positive sign, like this is 2, now I have to see
which sign I have to put in front of 2.
So, first of all we will see its address.
Means, in which row and which column it is.
We will see that.
In which row 2 is? In row 1.
And in which column it is? In column 1.
So, row 1, column 1.
This is its address.
Now, what to do with its row number and column number?
Add it.
How to write its address? I told you.
We used to write it as aij.
Where i is row number and j is column number.
So, i plus j.
So, its address is a11.
Address is a11.
So, row 1 and column 1.
Add both of them.
1 plus 1.
Now you have 2.
Now, whatever row number and column number you get by
adding, you will see that.
If it is even, then you will put plus sign.
That is, if i plus j is even, then you will put plus sign
in front of that entry.
Otherwise, if it is odd, then what will be in front of that
entry?
Minus sign.
Let's see row 2.
Its address was a11.
So, 1 plus 1.
2.
That is even number.
So, what will be the sign in front of 2?
Plus sign.
Now, let's go to row 1.
After 2, 1 is written.
Look at the address of 1.
Row 1.
So, its address is row number 1.
And what is the column number?
C2.
So, column 2.
Its address is a12.
Do 1 plus 2.
1 plus 2.
That is 3.
Which is odd number.
So, what will be in front of this?
Negative sign.
Let's see the corresponding of 4.
What is the address of 4?
Row 1.
Column 3.
That is a13.
So, do 1 plus 3 in a13.
That is 4, which is even number.
So, this will be plus.
So, what did you learn? What will be the sign in front of
this?
That is what you learned.
So, you can remember directly.
That in front of 1, it will be plus.
In front of 2, it will be minus.
Alternatively.
And then, in front of this, it will be plus.
And the reason for their being plus and minus.
I have already explained to you.
So, what did you do first?
That any row or column.
All three elements of it.
You have to space them and write.
Then, what did you do in the second thing?
In the second thing, you have to put their signs in front
of them.
That in front of them, it will be plus or minus.
You have to put that individually.
Now, what is the third thing that you have to do?
You have to do this.
From their small determinant.
Into.
What do I mean by small determinant? We can also call it
minor.
But, I have not taught you the word minor yet.
So, I am saying determinant.
What do you mean by small determinant? This matrix is of
3x3 order.
Which matrix will have a smaller determinant than this? 2x2
order.
So, we have to cut down this 3x3.
In 2x2 determinant.
Because, we have learned that.
So, let us see how we will cut down this.
So, first of all.
Which determinant will multiply with 2? Let us see that.
We took 2.
Whatever row 2 is coming in.
What do you do to that?
Delete it.
So, 2 is coming in our first row.
So, I am deleting first row.
I have deleted first row.
Okay.
And, which column is coming in?
In first column.
So, I have deleted first column also.
The remaining entries.
The remaining entries will be 4.
So, what is left?
These 4 are left.
3, 2, 5 and 6.
When we have deleted these 2 rows and columns.
So, we have these 4 entries left.
What do you do to them?
Line wise.
In the order in which they are written.
In the same order.
Copy paste them here.
Done.
So, see.
We asked for 3x3 determinant.
After cutting down.
Which one is coming?
In 2x2.
We have to solve this.
We have just learned in the first one.
Then, we will solve this in that way.
Easy.
Then, let us see.
What will multiply with 1?
We have to do the same with 1.
In the row in which 1 lies.
Cancel that row.
In the first row, 1 lies in this row.
So, we have cancelled it.
So, remove this.
We have removed this.
And, which column lies in the second column? We have
removed that also.
Now, what is left with us?
What is left?
Minus 2, 1, 2, 6.
We will write that line wise.
First, I will bring this back.
Then, we will write.
What is left?
Minus 2, 1, 2, 6.
I hope, it will be clear.
If it is clear, then try once quickly.
Which determinant will you multiply with 4?
Practice it quickly.
See.
For 4, which row will we delete? In this row, we have
deleted this row.
In this column, we have deleted this column.
Which are the remaining entries?
Minus 2, 1, 3, 5.
Copy them in the same order.
Then, you will get the determinant of this also.
What is this?
Minus 2, 1, 3, 5.
These were the three steps.
First, take any row.
Write its elements.
In the second step, put signs.
In the third step, multiply the determinant of the smaller
row.
Or, multiply the minor row.
We have done that.
Now, we can solve this easily.
How to solve the indeterminant?
By cross multiplying.
Multiply the denominator.
2 into 3 into 6 is equal to 18.
Minus 10.
Minus 1.
Minus 12.
Minus 2 into 1 is equal to 2.
Plus 4 into minus 10 is equal to minus.
Solve it.
8 into 2 is equal to 16.
Minus minus is equal to plus.
14.
Minus 13 into 4 is equal to 12.
Solve it.
Minus 22 is equal to 0.
I hope you have understood.
As soon as I taught you to expand along row 1, expand along
any other row or column.
Your answer will be the same.
That is minus 22.
If this is correct, your answer will be the same.
Expand along any row or column.
You just have to keep in mind, what will be the
corresponding minor multiply in front of that entry. And
what will be the sign in front of that entry.
That is the main thing.
I hope you have understood the determinant.
All three matrices are not asked.
Mostly, 3x3 and 2x2 are asked.
So, we have done that.
So, keep practicing this.
Keep asking questions related to this.
See you in the next video.
Till then, thank you for watching the video.
Discounts or even cash rewards.
Count me in.
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classmate