Next we will do multiplication of matrices.
Now if we multiply any two matrices, then we will have two
cases in multiplication.
Let's do multiplication of matrices.
In multiplication of matrices, we will have two cases.
Multiplication.
The third operation that we are applying in matrices, we
are applying multiplication.
There will be two cases in this.
One will be scalar multiplication.
And one will be normal matrix multiplication.
What is scalar?
Scalar means,
You have seen scalar matrix.
What is the meaning of scalar word in that?
Same constant number was in the whole diagonal.
So what is the meaning of same scalar?
Any constant number, or any other same alphabet, whatever
you are multiplying, multiply it with every element of the
whole matrix.
Suppose you are given, alpha into A matrix.
And A matrix is given as 2, minus 1, 3, and minus 2.
This is your A matrix.
Now you have to multiply alpha.
So what will you do with alpha?
Individually, multiply it with every element.
So this will become,
Alpha will become 2 alpha.
Minus 1 will become minus alpha.
3 will become 3 alpha.
And this will become minus 2 alpha.
So this is our resultant matrix.
So scalar multiplication is very easy.
You have to multiply it with every element.
You can give 1, 5, 10, alpha, x, y, z.
You have to multiply every element individually.
Now the main and most difficult, addition, subtraction,
scalar multiplication, all 12th grade students can do.
But those who are not able to do, those who face
difficulty, that is matrix multiplication.
Multiplying 2 matrices.
So just now we wrote the condition of subtraction and
addition, and also the condition of equality.
In those conditions, the order of the matrices should be
same.
But in this, suppose we have A matrix, and we are
multiplying B matrix in it.
Will there be any order of A?
Suppose we have order of A.
The order of A is m x n.
That means it has m rows and n columns.
The order of B is r x c.
B has r rows and c columns.
So what will be the condition in this?
That 2 matrices can be multiplied less.
What is the condition of multiplication of matrix?
The order will matter in this.
For example, if there are numbers, write 2 x 3, or 3 x 6,
write 2.
Both will have same answer.
This is commutative.
So write in front and back.
Order does not matter here.
But when we talk about matrix multiplication, in that,
order will matter.
Order means placement of matrix.
Who wrote the first matrix?
Which matrix is written later?
That will matter.
So if we have A matrix in front, and B matrix in back,
whose order is m x n for A, and r x c for B.
So what will be the condition of this?
The number of columns of first matrix, should be equal to
the number of rows of second matrix.
If n is equal to r, means the number of columns of first
matrix, should be equal to the number of rows of second
matrix.
Only in this particular case, multiplication will be
possible.
Otherwise, it will not be possible.
Otherwise, we will not be able to multiply.
So let us understand this with an example.
Write the question.
We have A matrix, and B matrix.
We have to subtract A x B matrix.
So first of all, in multiplication, we will write the order
of both.
Then we will check the number of columns of first matrix,
and the number of rows of second matrix.
So what will be the size of A?
Write.
What will be the size of A?
There is one row and two columns.
So what will be the size of A?
A will be 1 row and 2 columns.
1 x 2.
What will be the size of B?
B has 2 rows and 2 columns.
So the size of B will be 2 x 2.
Now check the number of columns of B.
It is 2.
The number of columns of B is 2.
And the number of rows of B is 2.
So both are equal.
So what does this imply?
This implies that multiplication is possible.
If both of them were not equal, then we would have directly
said that multiplication does not hold.
Multiplication of both of these matrices cannot be done.
Now see, first we checked the condition.
We took out the order and checked the number of rows and
columns.
We fulfilled the condition.
Now moving on to the next step.
Now we will multiply.
Will we do the same as we did in A plus B?
We will do the sum of the same address.
We cannot do that.
Because A has only 2 elements.
B has 4 elements.
So we cannot do the component wise multiplication.
Because A has less rows and B has more rows.
So what we will do is, there is a different way to solve
this.
I will rub this and tell you.
You must have checked the condition and noted it down.
Now we will tell you how to solve it.
I will write the question again.
What was your A matrix? The elements can change.
What is your B matrix?
Suppose there are 4 elements.
I will write anything.
Now how to multiply it? We were taking out A into B.
How to take out A into B? First we will check the row of A.
We marked the row of A.
We will check the 1st row.
We will check the 1st column of B.
We will check the 1st row of A.
We will check the 1st column of B.
We checked that 1st row of A is 1, 2.
And 1st column of B is 3, 0.
Now multiply 1st column of A with 1st column of B.
Multiply 1 with 3.
So 1 times 3.
We will mark plus sign on the middle.
and multiply the second one by second that is 2 into 0 so
this is your first entry entry at A11 address now how to
get second? what will be the element of A11?
A12 means first row and second column so we will leave A as
first row now for second column, what will we do for B? we
will see first column of A and second column of B so what
will happen? multiply first one by first one so we will
multiply 1 by 1 minus
2 and multiply it with plus sign and multiply 2 with 4. Now
what should have come here?
Here, below will come a 2 1 2nd row 1st column but if we
look at a, b, c, d, e, f, g, h, we were looking at rows.
We were looking at rows of A and columns of B.
So, there is no second row of A. It means that we do not
have it in front.
We will get the resultant matrix by solving only this much.
So, what will be the result of solving this matrix?
This would be 1 into 3 plus 2 is equal to 3 and minus 2
plus 8 is equal to 6.
So, this will be our resultant matrix.
And what is the order of this? 1 cross 2. Did everyone
understand the order of 1 cross 2 and how to multiply?
I hope you will understand this.
And you practice more questions related to this.
Then we will bring some hard questions in the next class.
Thank you for watching the video.
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