in-exercises-11-and-12-use-an-inverse-matrix-to-find-mathbfx_mathcalb-for-the-given-mathbfx-and-ma-2

Question

In Exercises 11 and $12,$ use an inverse matrix to find $[\mathbf{x}]_{\mathcal{B}}$ for the given $\mathbf{x}$ and $\mathcal{B} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{l}{4} \ {5}\end{array}ight],\left[\begin{array}{l}{6} \ {7}\end{array}ight]ight\}, \mathbf{x}=\left[\begin{array}{l}{2} \ {0}\end{array}ight]
$$

Doruk Isik · Accepted Answer

in this problem. We&#x27;re given a basis in defectors that form the base is so we have B one and B two and were given an expected and we&#x27;re has to find the coordinate director. So we know that X is equal to P. B, sir. Change important methods. But like by the court in Vector and from this, we can say that then the coordinate factor is equity universal PB times multiplied by a factor X What&#x27;s PB PB still lamented metrics formed by defectors that for at basis beat So it is or fire and 67 So from this we find P being worse to be negative seven over to three five over to the naked too. And to find the coordinate factor, we&#x27;re gonna both fly pt nurse with the given X vector and refined according it&#x27;s better to be negative. Seven Fine

Runpeng Li · Answer

Okay, So in this problem, we only need to apply our the&#x27;re, um your mate, you know, textbook. So it says if we have the Matrix P Well, in this case, our matrix peas. Just a span, not B one B two. So Matrix B. We know this matrix. Then our bi matrix will be peeing burst. Time say times P. So we just need to find out the product. So, Pete, members, by my calculation, I won&#x27;t do this step by step. Just a house. Just write down this of embers. It&#x27;s 11 I think the 23 now Matrix A is next to one for next 2 to 3 and matrix P. It&#x27;s just a three negative 1 to 1. Okay, So as first find, it was too part of the first product. So first entry, Nick one times. Like to warm. Plus two two times one. So that&#x27;s negative. Three and one plus one times four plus one times three. So that&#x27;s seven and two minus six. So it will be negative for And the last entry we have connective aid. That&#x27;s nice. So it&#x27;s one now, three negative. 1 to 1. Okay. Now for this to matrix. We&#x27;ll search for these two missions, since we have next of three times three active and I last 14. So it will be five and three. Ah, seven. So he&#x27;s 10 and next to 12. Us, too. So selective. 10. And we have four plus one. So he&#x27;s five. No, just put the skater into our matrix. So we have one too connected to and one So this is

Anthony Pisaturo · Answer

So this is problem 12 Section 7.7 on were asked to show that matrix A is the urn bursts of matrix speak. And to go ahead and prove that we have to show that a matrix a times matrix B is equal to matrix B times matrix A which is also equal to the identity matrix. Um and so the first thing we&#x27;ll do is go ahead and dio a B on and on another page, I went ahead and showed out the work. So this is matrix a, uh, times matrix B, and then this. If you can kind of follow what&#x27;s happening here, um, comes out two the identity matrix and then from here, um, I need to do B A. So be times, eh? I went ahead and typed it out as well. Um, and this is also gonna be equal to the identity matrix. And so because in our case, a B equals B A equals the identity matrix, we can say, yes, they are the inverse of each other

Aymara Gallardo · Answer

okay. We have to prove that these two Matrix A and B are members of each other. For that, multiply them. Oh, in theory, to program their immerse of each other, we have to move applied from both sides. But in the case of a square matrix, um, if you seem burst from one side will be from the other. So let&#x27;s go ahead, Mrs a six and multiply them. We&#x27;re on the side. You want? Okay, so the scholar can go off. Ron, let&#x27;s go ahead and make the most obligation. Or here. Let&#x27;s see, we have six times. One, that six. We have 17 things, too. So that is 34 and we have 12 times three. That&#x27;s 36. That&#x27;s for the 1st 1 Here. We have six times for that. He&#x27;s 24 17 times. Cyril, that Cyril Miners 12 times, too. That is 24 minus 24 on the bottom. We have six times. One that six, 17 times six and 12 times nine. Okay, it&#x27;s prepare the calculator. Make these numbers. Let me continue over here. Okay, so we have Cyril Times one. We have negative six. We&#x27;re doing two times negative. Three and then three times to that&#x27;s six. In the next column, we have Cyril Cyril four times to That&#x27;s Hey, and then we have the last column that is cereal negative. 12 Negative. 18. Two times nine Daddy&#x27;s So let me see. Um, this is wrong. He&#x27;s zero cereal and we have two times. Two. That&#x27;s four. Okay, how does this go ahead with the last one again? Have Cyril Times one that Cyril. Negative. 18. And then we have two times nine. That&#x27;s 18. Okay. And the last one, The last column. We have negative, too. Since one. That&#x27;s negative, too. We have negative 10 and we can 12. We have negative eight. We have zero, and we have eight. And for the last one, we have negative, too. Negative. Um 30 6 30 and negative. And positive. 36. Okay, lout. Now, let&#x27;s go ahead and make this calculation What we have lived from the 1st 1 piece for you have 24 minus 24. That sirrah here. If you make the calculation, this is Sierra. You&#x27;re a zero 40 here, Cyril. On the first ones here on the 2nd 1 and four. So that East 10 Cyril Cyril one Sarah zero Cyril one

Noah Duncan · Answer

Okay, So in this problem were given this matrix A here, which is too negative, 61 negative too. And we need to find a inverse using this special formula that were given. And then we need to verify that, um, a inverse is correct by multiplying a times a inverse, innate, inverse times A. And in both of those scenarios, we should get the identity matrix back. So we&#x27;re gonna do this step by step. So to get the to get a inverse there&#x27;s a special formula. So it&#x27;s gonna be one over a D, which is negative two times too. So that&#x27;s negative. Four minus BC, which is negative six times one. So it&#x27;s gonna be minus negative six, in other words, plus six. And then we&#x27;re gonna multiply that fraction, whatever it comes out to be by this new matrix where we flip the things on this diagonal and we negate the things on this diagonal. So if we simplify this matrix out, we multiply This fraction, which negative four plus six here is gonna give us just positive too. So that&#x27;s the 1/2 times each of these entries in this matrix. Tell me 1/2 times negative to just negative one me 1/2 time, six. Just three. It&#x27;s gonna be 1/2 times negative one, which is negative. 1/2. And it&#x27;s gonna be 1/2 times too, which is one. So so far, this is looking like our candidate for a inverse. We need to just verify that it works by multiplying it by a and by multiplying a buy it in order to see if it gives us the correct results. So to do that, we&#x27;re gonna take a, um, which has a negative two right here. Were gonna multiply that by a inverse and multiplying this. We should get the two by two identity matrix back. So let&#x27;s see if that happens. Take the first row, multiply it by the first column here, see what we get. So it&#x27;s gonna be two times negative one, which is negative, too. And they&#x27;re gonna add that to negative six times 1/2 times negative 1/2 which is three. So it&#x27;s negative. Two plus three, um, which is one. So we&#x27;ll get there, multiply that first row tends a second column That&#x27;s gonna be two times three, which is six plus negative. Six times one, which is negative. Six. That&#x27;s six minus six, which is zero. So we get there. Don&#x27;t take the second round. Do the same thing. Second row tempts The first column is gonna be one times negative one, which is negative. One plus negative two turns negative half, which is positive one. That&#x27;s negative. One plus one, which is zero. So we&#x27;re good there and then we&#x27;re gonna take the second row multiplied by the second column. It&#x27;s gonna be easier one times three, which is three plus negative. Too tense one which is negative too. So it&#x27;s three minus two, which is one. So we get there and we need to check. Ah, tohave them flip flopped and check that it works that way, too, because we&#x27;re not here in Theed that that always works. Interest is don&#x27;t always work out that way. So you need to verify the day. Indeed. Do give what we want, which is the two by two identity matrix. Good to see an exact process. Take the first row multiplied by each column one by one. So it&#x27;s gonna be negative. One tends to, which is negative too, plus three times one which is three. So it&#x27;s negative. Two plus three, which is one. So we&#x27;ll get there. It&#x27;s a good sign. Were to take the first road times a second column B negative one tends negative six, which is positive. Six plus three times negative, too, which is negative. Six. That&#x27;s six Minuses, which is zero great sign. Take the second road in the same exact thing. It&#x27;s the negative. 1/2 times, too, which is negative. One plus one times one, which is one. So it&#x27;s negative one plus one, which is zero. I&#x27;m going to take a second room multiplied by the second column. Negative 1/2 times negative. Six. It&#x27;s gonna be positive. Three plus, um, one times negative, too, which is negative, too. So it&#x27;s three plus negative, too, which is one. So we see that in both cases we get the multiplication live. Uh, are we get the identity matrix, the two by two identity matrix back. So you&#x27;re

Aymara Gallardo · Answer

Okay, so we have to prove that the Matrix A is a merciful being. So we have to protect a times B is identity and the other way around. So let&#x27;s make that brother first. That&#x27;s a Now be okay. So we kept negative Two friends. One that&#x27;s negative. One class three have stems to this three. So they keep us three minus two Now on the bottom, which have negative six, three cups, things for the makes six and end up here. We have one minus one on the bottom. We shall city minus two. And that is exactly then ended metrics. Okay, this calculate now be James a See if you can fit in here. Any of you to 13 Perhaps nudity won&#x27;t have. Where to three. What? Okay, what do we have? It&#x27;s gonna be the first coordinate. Negative to last three in the second quarter, earnings get gonna have. Did he have minus three halves? We have here negative for less poor and on the bottom. We have six over too. Rhinos for over two you see in the first quarter than it would have one in the bottom. We have sirrah, Sirrah over here. And now this is too over to. That&#x27;s actually one. And so we&#x27;ve verified then, ese. Remember Soapy

Problem

The set $\mathcal{B}=\left\{1+t^{2}, t+t^{2}, 1+2…

Problem 12 Easy Difficulty

Answer

$\left[\begin{array}{c}{-7} \\ {5}\end{array}\right]$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Kristen K.

Vectors Intro

Vector Basics - Example 1

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$\left[\begin{array}{c}{-7} \\ {5}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Kristen K.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications