 Next property which is going to be an important one is look at the very first set of properties, that means any square, the sum of a symmetric and a skew symmetric matrix very similar to the property that we have learned in functions that any function can be written as the sum of an even and a lot of functions. See the analogy that is between matrices and the numbers themselves, I think I have not done that concern with you but I will soon be doing it in functions chapter that any function can be written as the sum of an even function and a lot of functions. But the simple way any square matrix can be broken down as a sum of a symmetric and a skew symmetric matrix and this is going to be one of the prospective questions for you in your school test and exams where they will give you a matrix A and you will ask you write it down as a sum of a symmetric and a skew symmetric matrix. So, will you be able to do that? Let us take a small example question based on this express the matrix A and skew symmetric matrices. Fourth property I know if A is a symmetric matrix then raising A to the power of will all be symmetric or cannot say that depends upon A. What do you call it? Symmetric input which I will show you. So, if A is symmetric it implies A transpose is A. Now we have heard that this is a transpose A. It is as per as transpose A. Please note any matrix if you are raising it to the power of N by the way it has to be a square matrix for that right. Why I am saying that? Because if it is a square matrix you can also do correct. If this is m cross N this is also m cross N I will say multiplication cannot happen correct. So, when you are raising a matrix to an integral power or positive integral power you can only do it when the matrix is a square matrix. So, we know this property and you know from this that A transpose is going to be A back again that means transpose A to the power N gives you A to the power N which means A to the power N will also be symmetric. Is that right? If A to the power N what do you mean? Symmetric input is an odd difference. Is that right? How do we show this? Very simple. If let's say A to the power N transpose according to the property it is A transpose to the power N and you know that A transpose is negative A. Now if your N is positive sorry if N is even it will give you back to the power N. In that case the result would become a symmetric matrix. Is that right? Because you are starting from this and you are ending from this and this will always only happen very easily. That means it will become a skew symmetric matrix where N is if you have done inverse of a matrix in school at least. So, I will come to the end. So, let's take questions based on this. Let's see what the proportions are playing on the views of these properties. Question number one if A and B are symmetric I will just write a short form for symmetric and it is known that A B is B A and it is known that A B is B A then take symmetric option B skew symmetric option D none of the above. By the way I would like I think the property of transpose will help us in this. So, if you do A inverse transpose it is as good as A transpose inverse that means you are back to A inverse. So, you start with A inverse transpose and you are back to A inverse. What does it mean? A inverse will also be symmetric. What is symmetric? Very easy question. So, you just have to do that on this. What do you get? This is going to be A transpose inverse which is going to be A inverse into this term or negative of this term. So, what should I do in this case? A inverse is symmetric. Symmetry A inverse is symmetric. What do we do each to this value? What do we do with it? Have I used this property yet? Can I use it? A B will be symmetric. A B is symmetric. How do you show that this is symmetric? A inverse is symmetric. B is also symmetric. If A B is equal to B N. No, no, no. You are not getting my question. The question is there are some matrix you are asking. If you claim that this is symmetric you should get back the same matrix over here. So, multiply by A both sides. You can identify matrix. You become PI. You become PI. Yeah. See, what are you doing? I have to do this somehow that this and this are same, right? Okay. Let me start with LHS. A inverse B we have. Can I do one thing first? Can I write this with pre-facto A? Pre-facto A means multiplying before. So, this also has to be multiplying with pre-facto A. So, A inverse is known to be I. So, I get a B correct. Now, here we know that A B and B are same. So, can I flip the position of this? Can I make this A B as we can multiply these two together first? Which is again I. You know, B I is B only just like here I B is equal to this. So, basically B is equal to B which means the assumption that A inverse B and B inverse A are same is actually correct. So, you can put this back as A inverse B and therefore, transposedly the matrix giving you the same matrix back implies that matrix was symmetric. Okay. Now, these are the properties which are helping us to solve this question. We never assume any matrix to be something. We solve it just by the use of properties. Let's take another one. Next is A B and A plus B are non-singular matrices. Now, by the way, we have not officially learned what a singular non-singular matrix is. Non-singular matrix is basically a matrix whose determinant or whose inverse can be found out. Okay. That means it is invertible. Then A into A plus B inverse into B whole inverse. We have not learned inverse property. Wait one second. Sorry about that. We have not learned inverse property. Yes, B is given by 2.5 minus half, 3 by 2. 1, 1, 0, 1. So, A is given by 1, 1, 0, 1. And 2 is given by P and P transpose. And X is given by P transpose 2 to the power 2,000 times P. Then, find the matrix X to the matrix. Just like A inverse is I, A inverse A is also I. That's all you need to know is A A inverse and A inverse A whole inverse. Okay. I know many of you have no clue how to solve this. First of all, I would like to know from you. If I say Q to the power 2005, what does it mean? Q into Q into Q into Q, how many times? 2,000 times. That means you have to write P A P transpose, P A P transpose, P A P transpose, P A P transpose almost 2,500. Correct? Now, when you see, I would like all of you to tell me what will happen if you multiply P transpose into P. All of you please do this operation P transpose transpose into P minus P square minus G square minus P square. What is the P square? Can you simplify it further? What are the values? If I had to write P square only, I would not have mentioned what is P at all. I am just telling you the process. You don't have to think anything. Just do as I say. Just do P transpose into P. Tell me the answer. It is an identity matrix. 2 3 by 4 is Z 3 by 4 minus 2 3 by 4 is 0. 1 by 4 plus 3 by 4 is 1. So, can I say this 2 term could be clubbed as I. Similarly, here this could be clubbed as I. I will like that and say before this there should be clubbed as I. So, do you see that even this club is something and become I, even this club is something and become I. So, can I say this entire expression 2 to the power 2005 is nothing but P A to the power of 2005 into P transpose. There is something very special in A. A to the power of 2005 can be found out. Just try doing A square A cube will automatically see the power. Just do A square. A square means, so this will be 1, this will be 2, this will be 0 and this will be 1 again. A cube means A square into A. So, just try multiplying this with A again 1 1 0 1. So, this will be 1, this will be 3, this will be 0, this will be 1 again. Do you see that? Only this guy is changing correct. So, if you continue the line I can say A to the power 2005 will be 1 0 2005 A to the power 2005. Yes sir. Now, the question was finding X. X itself was P transpose P A to the power 2005 into P transpose. Into P. Now, again these two gentlemen will become I I and that means A to the power 2005 itself is your answer. So, this becomes your answer. Isn't it? Q was actually this, Q was actually what did we found out? Absolutely. Yeah, we found out this part. This was A to the power 1, this was this term correct. Now, this here to 3 multiplied by 8 back into P. So, P transpose P is back to I again and P transpose into P is back to I. So, it is I A to the power 2005 I which is A to the power 2005. So, whatever you got for 8 to the power 2005 that will be your answer.