 Now, let us talk about inverse of a matrix that should be very important that is a first thing. But before that we have to talk about adjoint, adjoint of a matrix that adjoint of a matrix can be found only for square matrix and what does this concept adjoint of a matrix. Now let us say if I take any matrix let us say matrix A let us take a matrix A of order 3 means that they are co-factors. Now what is co-factors? Let us see this concept very very carefully. Co-factors of a matrix a co-factor of an element let us say I want to find co-factor of A11 then what do we do? We do minus 1 to the power of 1 plus 1 why this 1 plus 1? Because A11 belongs to first row first column. Into minor of A11 not minor. Minor means minor of A11. Minor means hide the column and the row in which A11 falls. Whatever number you see let us say it is only 1 number left then that number could be the minor. But if you see 4 numbers like this then do a cross multiplication just like you do in the current. So it will be A22 minus A3 A22 into A33 minus A32 into A32 that would be called the minor. So co-factor of that element is nothing but minor of that element along with the sign like this. So in general I can say co-factor of Aij will be minus 1 to the power i minus j minor of Aij. Then whatever 9 elements you see you will find the determinant of it. So minor is nothing but the determinant of the matrix that you get or the sub matrix that you get after hiding the row in the column which that element falls. So Sheeran was asking the question had there been a 4 by 4 what would have been the minor of A11? 9 elements. There would be 9 elements over here left? Determinant of minor with the sign is the co-factor. Is that fine? Now let us say the co-factor in the matrix C like this. It is not probably the right co-factor matrix as COS bracket A that is co-factor of A. So A has been given to you if you make a matrix by replacing the elements with the respective co-factors the matrix obtained is called the co-factor matrix. If you transpose the co-factor matrix then you get something called the adjoint of A. So if you transpose this you get the adjoint of A. Is that fine? Understanding adjoint is very important before we understand the inverse of a matrix because we can actually find the inverse of a matrix from its adjoint. I will come to it in little later on time. In the same position that is why I said replace the elements with their co-factor. So A11 co-factor would be replacing A11. A12 co-factor would be replacing A11. Now I will give you a question. Find the adjoint of A if you are A13-52400 minus P6-1. This exercise together, so you are revising class A11 simultaneously. This exercise you are revising class A11. Then math theory. Two topics would be exactly the same. Let's talk about what is the co-factor for 1. 6 into 0 is 0. 4 minus 0 is 4. So I am writing co-factor 4. Remember 1 to the power 2. So there is no sign along with 4. What about 3? First of all 3, the sign attached to it is minus. Then hide the column and the row in this 3-4. So it will be 2, so it will be minus 2. Take your time. If you have not understood this, please ask me. Next, for minus 5, the sign would be positive again. So I am going to write everything. So hide the column, hide the row. What do you see? You see these correlations. 12 minus minus 12. That is 24. Is that fine? Next. For 2 again the sign would be negative. Hide the column, hide the row and the column which it falls. So you have 3 minus 5, 6 and 1. So 3 minus 30, 33. So altogether the answer would be minus 30. And 33 is the minus for the sign. So minus sign into minus will become your co-factor. 4 comes in 2nd to 2nd column. So sign will be positive. So hide this. So it will be minus 14. For 0, again 0 comes in the 2nd to 3rd column. So you will have minus sign already. So hide the column, hide the row. So you see 6 plus 9 which is 15. Please understand this process of finding co-factor matrix very very clearly. Next. Minus 3. Again plus sign. Hide the row, hide the column so you see plus 20. 0 minus minus 20. Clear? No. How does that? The negative sign of the number doesn't matter. The negative sign attached to its position matters. I plus minus 1 to the power minus j is the sign. Minus 3. Minus 3 in the sign would be minus 1 to the power 3 plus 1. It belongs to the third row first column. So that's the positive. So I didn't write any sign already. Hide this 3 to 0 minus 4 into minus 5. So minus of minus 20. Then for this it will be minus sign again. So hide this, hide this. It will be plus 10 which means minus into plus 10 which is minus. For this again a plus sign. Minus 2. Plus minus plus minus 6. Minus. So it will be hide this, hide this 4 minus 6 which is minus 2. Yes or no? Not sometimes. Many a times it leads to a lot of silly mistakes. People do mistakes, normally do things. She has calculating, doing the determinant minus makes that happen. So for a 3 most commonly asked matrices for you to find the adjoint. I will tell you a shorter way to get the go factor matrix and hence the adjoint. By the way if I find the adjoint here it's very simple. What I have to do? Just transpose it. So the answer will be 4 minus 2, 24, minus 10 and 3, minus 14, minus 15, 20, minus 10, minus 2. Now I tell you a shorter way to find out the go factor and hence the adjoint in a much faster way. But again don't use it in school very explicitly. You may use it just at the upper part to test your result. Let's say I want to find out the go factor matrix for this matrix. Ok go factor matrix for this matrix. So what I will do? I will first write down the elements as they have been given in the matrix itself. Ends as I have shown you by the, no minus 2 will be minus 2, 80, 12 plus 12 would be 24, minus 70, minus 3 will be minus 33, 1 minus 15 will be minus 14, minus 9, minus 6 will be minus 15, 0, minus 20 will be plus 20, minus 10, minus 0 will be this, 4 minus 6 will be minus 2, your go factor is 80, no need. Yeah sure. You have to say 1 to 3. It's like 1 is all, 1 is 3. Yeah specifically to 3 to 3. This specific to 3 to 3, so I have to say that but yes. Specific to 3 to 3. Yes, you are shifting it again. I have to delete it, you are asking. I have to repeat one thing. Let's take another example. You have 2. Ok. So let's say I have another adjoint of, let's say 3 minus 2, 5, 1, minus 1, minus 6, 7, 2, 5. Now, again I am repeating the process. First copy down the same as such, so 3 minus 2, 5, 1 minus 1, minus 6, 7, 2, 5. Ok. Then copy the first row below the, this, copy the second row, so far so good. Now, copy the second column over here. Is that right? Now listen to this. First row completely, erase the first column completely. Now multiplication, whatever you do in determinants. So this will be minus 5. So, co-factor of a would be plus 12, 7, minus 42, minus 5, minus 47. Then 2 plus 7, 9, 10 plus 10, 20, 15 minus 35, minus 20, minus 14, minus 6, minus 20 again. Then 12 plus 5, 17, then 5 plus 18, 23. Then what do you get? Minus 3 plus 2. Is that right? The transpose of whatever you have. Perforate minus 2, what? Isn't it? Now the model of the story is exchange the position of these two and change their, correct? Exchange and change their signs. That means it was minus 6, 5, minus 2. What is the minus 2, 3, 6, minus 5? 4 by 4, maybe you will not be asked also. Very rare. What is actually I joined helpfully? Co-factor checking is a waste of time because almost 30 students will be there in your class. 24. I don't think you should just check, sit and check for the line. Yeah, that's what I'm saying. She may only check the final design, the spectrum of how you have got it. What's also they say if you don't take a mistake is the answer, nobody knows. And you are worried so much, no? Just do it in the rough one. Okay, fine. Now you know the most important one. First of all, astonishing to us, a scalar magnifies this. Let me take a simple example to understand. Let's say A is 1, 3, 2, minus 4. Okay. I'm taking the 2 by 2 so that our working is very less. I would like also you to give me, what is the adjoint of A? Given that I already told you that there is minus 4, 1, minus 3. Okay. All of you perform this operation A into adjoint of A. So multiply A with the adjoint of A. No need to check this one because it has to be same. So 1, 3, 2, minus 4 multiplied with minus 4, minus 3, minus 2, 1. What do you get? Minus 4, minus 6, minus 10, minus 3, minus 3, plus 3, 0. Minus 10, 0, 0, minus. Minus 8, plus 8, 0. Minus 6, minus 4, minus 10. Do you see that? You can actually write it as minus 10 times 1, 0, 0, 1. It's just nothing but minus 10 times I2. And minus 10 you would be surprised to know that if you find the determinant of this matrix, that's minus 4, minus 6, which is minus 10. That's why we have this property that A into adjoint of A where A is, let's say, a square matrix of order n of order n. Then A into adjoint of A would be adjoint of A into A. And that product will be A determinant into I n. So basically this is which type of a matrix? This is a scalar matrix. Remember when we were doing types of matrix, I told you scalar matrix is a matrix where the diagonal elements are the same, non-zero scalar matrix. Now from this property comes out the concept of inverse of a matrix. We'll talk about it when I'm talking about inverse. As of now, this property, please note it up. Next, adjoint of AB follows the law of reversal. It's adjoint of B into adjoint of A. Again, law of reversal. A inverse, AB inverse is B inverse A inverse. We'll talk about it in something. Transposing the adjoint itself. Adjoint of A comes first. This as it has. Transposing the adjoint of AB itself.