Hello everyone, welcome to the new lecture of matrices.
In the last class, what did we study in matrices?
We studied different types of matrices.
Now, we will see that when are matrices equal?
When will any two matrices be equal?
So, if we are saying that any two things are equal, then in
what sense are we saying?
For example, if there are two triangles, when will we say
that they are equal?
When both the triangles have equal size.
Suppose, we have this triangle.
Then, the other triangle should also coincide with this
triangle.
All the vertices from all sides should be same.
Length should also be same.
So, the same thing happens in matrices.
When will two matrices be equal?
First of all, what should happen?
As we have seen that the size of the triangle should be
equal.
Similarly, we will see in matrices that what should be
equal to both the matrices?
Its size.
What is the matter of size?
It is the matter of number of rows and number of columns.
What should be the matter of both the matrices?
Number of rows should also be same.
And number of columns should also be equal.
So, first of all, we will see that what should be the
matter of both the matrices?
Size of both the matrices should be equal.
What should be the size of both the matrices?
It should be equal.
Now, if we have this matrix.
1, A, 3, 2
I am saying that this matrix is equal to any other matrix.
If I am saying that these two matrices are equal, then what
will happen? For example, what is the order of this matrix?
It is 2x2.
It has two rows and two columns.
So, what should be the order of this equal matrix?
It should be 2x2.
First of all, the order should be equal.
Then what should happen?
The element which is present at each place,
What should be the order of each address? It should be the
same element.
For example, this address is A11.
This is A11.
What is the address of this entry? It is A11.
So, the entry which is present at A11 should be equal to 1.
Suppose, here is X.
So, what will be the value of X?
If we are told that these two matrices are equal.
This is matrix A and this is matrix B.
So, what we will interpret from this?
The same element should be present at the same address.
If here is 1 at A11, then here also it should be 1.
If here is X, then what should be the value of X?
It should be 1.
Suppose, here is A at A22.
So, here also A will be present at A22.
Similarly, here is A at A21.
So, here also A will be present at A21.
That is 3.
So, here also it will be 3.
Here, what should be the value of A?
This is A22.
So, what should be the value of A22?
It should be 2.
Then, our matrix will be equal.
So, when will the two matrices be equal?
When their order or size should be same.
That is, the number of rows and columns should be equal.
And, the same element should be present at every address.
The same entry should be present.
Then, our two matrices will be equal.
Suppose, I give you a question.
See this question.
It is similar to this.
The question is,
X2, Here we have A matrix.
It is said that this is another matrix which is equal to B.
And, how should be B matrix?
Before writing the question, I am asking you.
What should be the order of B matrix?
If both the matrices are equal, then B should be equal to
this order.
What is the order of A?
How many rows? 1, 2.
How many columns? 1, 2, 3.
So, what is the order of A?
What is the size? 2 x 3.
So, the order of B matrix should also be 2 x 3.
That is, 2 rows and 3 columns.
So, this is the question.
This matrix equals to this matrix.
These two matrices are equal.
If they are not equal, then the elements of each address
will be equal.
So, what is on A11?
On A11, we have X.
On A11, we have minus 1.
So, what is the value of X? The value of X is minus 1.
On A12, we have 4.
On A12, we have 3.
On A12, we have 2.
Now, the value of Y should be equal to what?
What is the address of Y?
A22.
On A22, the address is 0.
So, the value of Y is 0.
Similarly, what is Z?
A23.
On A23, the address is 5.
So, the value of Z is 5.
Now you know when two matrices are equal and in what
condition they are equal.
Now after equal matrix we have operations on matrix.
What are the operations?
Like we do in numbers.
What are the operations in numbers?
Addition, subtraction, multiplication, division.
What will happen in matrix?
The operations will be addition, subtraction, scalar
multiplication and normal matrix multiplication.
There will be no division because we will not be able to
divide the matrix.
But we can add, subtract and multiply them.
So what will happen in operations on matrices?
Addition.
How will we add two matrices?
In this also there will be a condition that if we want to
add two matrices.
Like we have this one matrix.
1, 2.
In this we have typed wrongly.
Suppose we want to add 5 in both matrices.
We have wrong entry.
Now we want to add 5 in both matrices.
So what will we do?
We will add 5 in both matrices.
Because we have to change in all entries.
But if we want to write it in matrix then how will we do?
If we want to write plus 5 in both matrices then we will
write plus sign here.
And write 5 here.
Because we are writing plus 5 in this address.
And we are writing plus 5 in second address.
So we will write 5 here also.
Now we can add both matrices.
So what will happen?
Add 1 plus 5 component wise.
Add 1 from 5.
Add from same address.
1 plus 5 is 6.
2 plus 5 is 7.
This will be our matrix.
In this we have noticed one more thing.
Suppose we have this matrix.
5, 5.
And below also 2, 5.
2 cross 2.
So how will we add both of them?
In this we said add from same address.
But if we see here.
Below there are no elements.
So what will we add?
Children make mistake in this.
They take 0, 0.
If you increase or decrease any row or column in matrix.
Or add.
Then it will change in matrix.
Its order will change.
So we don't have that thing.
Which we already had.
We changed it.
So that is why we don't add it.
Whatever they do.
They add 0, 0.
And then they add this.
That becomes wrong.
So what will be the condition for addition? For addition
also same.
Same equality condition will be there.
Size or order of matrix should be same.
What should be there?
Size and order of matrix.
Or whatever matrix we are adding.
Size should be equal.
Then only it can be added.
If size is same.
Then what to do with the elements on same address?
Sum them together.
Now in the same way.
What will we get?
Case of subtraction.
Addition is done. Now subtraction. Subtraction or
difference of matrices.
There is no need to do anything in subtraction.
We just need to put a minus sign. This will also have the
same condition.
The condition will be same for both of them.
The order or size of the matrices on which we are operating
should be equal.
Only when the size is equal, we will be able to add or
subtract.
So, in subtraction also the size order should be same.
Suppose we have this matrix A matrix 1, 3, 6, 7.
We have to subtract this matrix 2, 2, 2, 2 from this matrix.
So, what we will do is, we will take the same address
element.
This is A11. This is also the other matrix.
This is the element of 11 address.
Write B11.
So, we will take the same address elements.
We will take the same address elements and keep the same
address P.
We will put the plus sign in the middle.
We will put the minus sign in the middle.
So, this will become 1.
Then we will put the minus sign in the middle.
And we will add 2 here.
This is A1.
This is the element of 11 address of the resultant matrix.
Similarly, this will become 3.
And this will operate with this.
This will become 3 minus 2.
This will become 6 and will operate with this 2.
This will become 6 minus 2.
This will become 7 and will operate with this 2.
That is 7 minus 2.
Now, solve this.
This is 1 minus 2 minus 1.
This is 1.
This is 4.
And 7 minus 2 is 5.
This is our resultant matrix.
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