Brass it through their 911 call it. The storm says Storm says the third given to Paulino mules, F and G. Then we have four statements than for any square matrix. Okay. No, it's and a scaler. Okay, jesus, you can say the N word. So well, we have that one F plus G of a equals F of a plus G. Ave dish, dish rag, suck my dialogue and then it's like years. So I'll prove this statement first rather than going through all of them. Oh yeah. Well what FB since is a polynomial? Am teach D. N. Yeah. Plus all the way down to a one T plus a zero. And I'm doing a version of G is a point A mule uh, B M teach the M and so on. All the way down to be won T plus B zero and just lack Yeah, black out. And therefore the value of F at our matrix A is A. M times A to the N and so on. All the way down to a one times A. Or say zero times I. And G. Of A. Yes, B M comes A to the M. And so on. All the way down to be one times A plus B zero times I Yes, adami Okay, now without loss of generality, suppose that end is less than or equals to end And let be I be equal to zero if I is greater than M. That's one of my favorites. What's that? Uh There follows that. F plus G. This is also a polynomial of the form A. M plus yeah, B M. T. To the end. All the way down to a one plus B. One T plus a zero plus B zero as a parent. And therefore it follows that F plus G. F. A. This is the polynomial A. N plus B, N times A. To the end and so on down to a one plus B. One times A plus a zero plus G zero times I. Yes. Then using distributive property, this is A. M ties eight to the end. He had to do the shooting plus B. M 80 M. And so on. All way down to a one times A plus D. One times A. Just 80 times I I plus B zero times I Yes. As soon as guys have said like yeah, getting out of the car boy George. And of course we can rewrite this as F. Of A plus G of A. But we wanted to show in part one in part two were asked to show that again for any square matrix A. In Scaler K. We had that F times G of A is five times G L. A. That video. It was well, once again by definition we have F times G is equal to C sub M plus M. T. To the N plus N. All the way down to C one T plus C zero. Where these coefficients? What are you talking? What's the dating shows that the guy at walmart are given by A zero BK plus a one Bkr miss one. All the way up to AK B0. Oh, but I thought you said the racist If you want to write this and collapsed form. Well, our function F times G is the sum from K equals zero to N plus M. Uh T to the cave. Uh C K times T to the Okay, this coefficient C K can be written as this. Um From I equals zero. Okay, A I B k minus I retaliation. And therefore by definition, F times G of A is equal to the sum from K equals zero to N plus M of C sub K. Thanks A to D K were eight to zero's I on the one hand. And on the other hand, five times G a day. By definition every Is the sum from I equals zero to M. Of ai times A to the eye, times the sum from jay equals zero to M. Of beasts of J times A two J. And then multiplying these sums, we get the some from I equals zero to end. Some from J equals 02 M. Of AIDS of I timepiece of J comes matrix A. To the I plus J power. Can we can rewrite this using the above notation as the sum from K equals zero. Although we have two N plus M of C sub K times A to the K. Which of course we recognize that the same as at times G of a. We have proven statement to as well. Shaun King knows immediately, like authorities are now saying as a white supremacist move ish Statement three says that K times f of a equals k times it is a there sean king, john king like a king. Yeah. Now, by definition are stellar. K times or function is the polynomial? Okay aside, then teach the end and so on. Falling down to K. A sub one T plus. Okay, A zero plastic. And so by definition we have the K times F of A is K times a seven times A to B. M. And so on. All the way down to K times A one times A plus K times a zero times the identity matrix. I factoring out. Okay, We get K times a seven 8 to the end, plus Down to a sub one times a plus zero times I recognize. This is the same as K times a definition F of a. This is what we wanted to prove for part three. Finally, statement four that that drive back says that feb 10 G of the product commutes. Yeah, it is. So this is equal to G L A times every day. And do the part Okay, Well, using a result in part two, We have that Guia Tom's F five is equal to G times F of a. Which we know from the properties of polynomial. This is equal to since polynomial commute. This is F times G of a. Which is then again by part 25 times G. They it's great like getting creepy voicemails.