(2) Consider the matrixA = | 3 5] Find the unique %,y e R, such that A2 IA + yI (where I represents the identity matrix) _...

If $\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] A\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, then find the matrix $A$.

We have fallen. # 14. Question says for metrics Equal to three fun. Okay so we have been provided with the metrics uh 312 and one. And we need to find the numbers A. And B. Such that A square place. What a square place A. Plans B. I. Equal to. In this case we need to find the value of A. And B. So let's get started. First of all, we have already done that is queries A. Into A. So 3121 In 23123121. These are the mattresses and will be multiplying like this. Three into 39 plus two and 21 two and three and 2 to six plus two. Then one into 3, 3 Plus one. And to place one. So it's quite becoming 11 8, four and 3. This is is square. Okay, now that is plaguing the value of A squared and A. In this equation. Given a question. That is the question. Number one, we'll be getting 11 8 for three. Bless A. And two A. Is given us 3211 Plus B. I will be out of order two x 2. So 10 is the Mathematics 10 01 equal to 0000 and zero. Okay. No We should be writing 11 8 for three less. three A. 2 a plus B zeros. It'll be equal to 000 and zero. Now let us add all things up. 11 plus three A plus B. Eight plus To a Plus zero. For players A plus zero and three players A plus B. This will be called to 0000. No comparing all the elements, we will be getting 11 plus three A Plus B equal to zero. An eighth place to a Play zero equal to zero. Yes. And 34 plus eight plus zero equal to zero and three plus eight. three Plus A Plus B. Okay. Scripless. Yeah A plus B equal to zero. Okay. Now from here we can get A call to -4. Okay? And if you plug in the value of A. Over here it will become three plus B equal to four. So B equals two. What? Okay. No. From this equation will be getting a. equal to -4. And if you plug in here in this equation be equal to Uh one. So both are satisfying the given condition. So we have answer equal to minus four and be equal to one. So this is answer. Thank you.

That you no ordered lying, are you? I am. I am Ernest Rightness. I dependant maybe get I am on this minus. Thinks I mean no. If I had a morsel, there is urgent metrics minus Okay. Yours is very cool. You are good. Play a six on this five. My leg in Venice. Oh, on simplifying this every day I am minus the inverse is equals Waas running, diverted, But it third way minus No. On Monday keeping eats any manner with over a six by minus one. But if we get I minus It was one of the very people food. I am looking it. When did you word? No Siegler.

Okay. Yeah. All right. We're gonna find the determinant of matrix A. By expanding this first column. So we're determined in the bay it's going to include three. It's going to include two. It's going to include one but we want to then look at their co factors. So the co factors are going to include a negative one for each of these but they're expert is going to be different. Three is in the first row, first column. So one plus one is two for the X men. Two is in the second row, first column. That's going to be three as the X moment. Then one is gonna have an expert before from there we want they're minor which is the two by two matrix of items that are not in the same row or column. So three is in the top row, it's in the first column. So 2 to 31 is going to be the two by two matrix there. Here we get 2331 as our matrix. and then one is going to give us 2322 sort of determinant of a is going to equal three times one times two minus six plus two times negative, one times two minus nine plus one times one times four minus six. So the determinant for this major because it's gonna be three times negative, four minus two times negative seven plus negative two. So we end up with the determinant Equalling zero, which means the inverse does not exist.

We were asked to find the null space of a The mission of the null space is given up top here, where the null space oven n by n Matrix A is a set of vectors x and are in such that the matrix a times the Doctor X is equal to the zero. The growing gives us the Matrix A is a three by three matrix shown here We can see that in equals. Three unions were in or three, which means our expect er's we're gonna be three by one virtues x one thanks to the next three. So now we can start solving this linear equation by reducing the matrix a tow reduced through a national informed. So we'll start that road, bro. Reduction blue. I've re written matrix A and I meant it form so we can start our row reduction and we can first move this term to zero. If we multiply a negative three fiber one and add that through to so writing that shorthand, we will multiply negative three by row one and added the road to that will give us minus three times one. So I'll rewrite our row. One that won't change. We have minus three times one plus three, which is zero minus three times I plus four I, which leaves us with all right and minus three times minus two plus or minus five. Which gives us one and my three times Europe list 00 in a row. Three states the same when it's one. When is three I I and zero. So now we can think this terms of zero. If we multiply row one about one and then just add that 03 and so we can write that shorthand is multiplying room one by one and adding it to three. So we have where one stays the same for two stays the same. Then we have adding row. 123 We have one minus 10 I minus three eyes minus two I and they in the minus to Plus I is I minus two in zero plus zero zero. Now we can we can multiply the second row, but to than that and that's a road three. Metal. Get rid of this term so we will multiply, wrote two by two and add the the three I was scrolling down. We have first Roach. There's the same one I minus two zero and multiplying right to by two. We have to time zero, which for third rule to time zero plus zero zero. Two times I plus a minus two I zero. So all right. At this second room, which there is the same I 10 then a to times one plus an I minus two leaves us with just and two time zero plus here is still zero. Now we can multiple I wrote to, but to and then and that's a real one. And then we can get rid of this term here, this term here. So we will be multiplying two by road to and ending it through one. So all right, out our third rooms, your zero i zero her second round will stay the same. 010 Now we're multiplying our second row by two and adding at the ones that we have to time, zero plus one is one hands a two times I plus eyes three, uh, in the two times it two times one plus a minus 20 And we have to time zero plus 00 And then we can now we can multiply Row three I I to get rid of the complex number. And also we can do the same thing with room to do so we'll do. Two things in the same operation will multiply Row three I I and multiply row to buy I So our first road stays the same for a second row is our time 005 times I which is minus one and hard times one, which is I and I think 00 when multiplying or three by I'm We have our time 00 I time 00 times eyes minus one at time 00 So we've reduced this to Rochel Informant weaken Soul We could start We can multiply our major cities together and sold this so rewrite are reduced matrix so are reduced matrix a and we're doing the a Times X which is equal to the zero factor. So we have our use matrix a one three I zero and then our second room zero minus one and I in our third row 00 minus one. Well, that times are expected, which is x one x two x three is equal to or zero Victor which is a three by 1000 so we can multiply this out one times x one three times x two and a zero times X three is equal to zero. We have a zero times x one minus, one times x two, which is minor sex, too. Plus five times X three equals zero. Then we have a zero times x 10 times x two The A minus one times x three sub minus x three is equal to zero. Well x three has to equal to zero has to be equal to zero. So that zero, which means X two has two equal zero, which means that X one has to equal zero. So we have one solution here, which is the zero victory so we can write our solution as the null space of a. So we have our extractor here, which we have one solution. So that means the null space of a is the zero victor. That's the answer