verify-the-statements-in-exercises-19-24-the-matrices-are-square-if-a-is-similar-to-b-then-a2-is-sim

Question

Verify the statements in Exercises $19-24 .$ The matrices are square.
If $A$ is similar to $B,$ then $A^{2}$ is similar to $B^{2}$

Runpeng Li · Accepted Answer

Okay, so for a problem, 20 were given the two to similar matrices. So by our definition of similar matrix, that means we There&#x27;s Derek. Six. There exists on the convertible matrix p such that a few times b times p m burst is a So this is our room assumption. Now we want to show that he squared is similar to be square now, To do that, we first consider a square to be a time saver. Now we substitute this expression of a So we have the PM&#x27;s be times P embers does be times. He&#x27;d have to be times p verse now because matrix multiplication is associative and we can calculate p first times peepers. So we have this and we kept with peace be times beavers then we who have we will have p times b squared times p yours. Now this shows that a squared is similar to

Runpeng Li · Answer

Okay, so for problem 22 were first given Matrix K, which is diagnosed Technol Izabal. So that means by definition we can have a have any murderball metrics say ass times as he first time says is De were where these are diving a matrix. Excuse me. Now also, by the definition off similarity then were given that be similar to a So we also have, um t inverse times. Hey, times t she&#x27;s bi. Of course, T has to be a murderer matrix. Now we first soft a by multiplying I&#x27;m on applying t from the left hand side of right hand side so that mrs, uh, sorry. Second question. We must have a which is tee times be I&#x27;m stealing verse. Now we put the D. C. Based compression into our first question. No, we have asked members times t hey, t inverse times asked. Okay, Now here. Uh, yeah, I I missed the Dagenham matrix D on the right hand side. Now here, uh, let&#x27;s take you to be a scene first times t so that, um our yeah, so our you immerse will be t t inverse times has. Okay, So since these two matrices are all in murderball, so you has to be in veritable so we can rewrite our expression here. D to be you times B times Ewing burst. So we&#x27;ve shown that be, by definition, bees.

Sriram Soundarrajan · Answer

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Pam Russell · Answer

So we&#x27;re gonna use direct multiplication to verify that the given to maitresse ease are in verses of one another. So if we have matrix A and we multiply it by a matrix, a inverse, if our result is the identity matrix that, in fact these two matrices are in verses of one another. So if we&#x27;re given Matrix A, which is a three by three matrix, and then we&#x27;re given a inverse three by three matrix If we multiply these two matrices together in our result is the three by three identity matrix that, in fact these two matrices are in verses of one another. So given a multiplied by a inverse, we&#x27;re gonna use matrix multiplication to verify that indeed, they are in verses of one another. So with a three by three, we&#x27;ll still use the dot product and we have three times eight plus five times negative five plus one times two three times negative 29 plus five times 19 plus one times negative eight and then we have three times three plus five times negative too, plus one times one one times a plus two times negative five plus one times two one times negative. 29. Close to times 19 plus one Times Negative eight one times three plus two times negative, too, plus one times one and then finally for row three. Two times a plus six times negative five plus seven times two two times Negative 29 plus six times 19 plus seven times Negative eight and then two times three plus six times negative, too, plus seven times one. Calculating all of these out in row one column one. We end up with a value of one in row One column to we get a zero and Row one column three. We also get a zero and wrote You column one. We get a zero in row two column to we get a one and in row two column three, you get a zero and finally in Row three column one. We get a zero and in row three column to we get a zero and in row three column three we get in one, and this is the three by three identity matrix. So we have verified that these two matrices are in verses of one another

Wesley Hines · Answer

So we&#x27;re trying to see if some properties that hold with equations and numbers hold with matrices also, and in this case, for your testing. If a B Square is equal to a squared B squared, and the thing about Major sees is that matrices can be a close enough to zero where a lot of multiplication properties start to fail. So, for example, if we have I&#x27;m interested. That&#x27;s pretty much one. We multiply it by a major See that is extremely close to being zero but is not quite zero. In this case, B Squared is going to be equal to the same matrix. Might be Square is going to be equal to the first road times. The first column, which will be one. And then we&#x27;ll, for every other column will be want to plane by a zero, so we&#x27;ll have 000 So, so far, so good. What about a squared? Well, a square every scenario. We&#x27;re going to have one plus one so a squared is actually going to become the two by two matrix of strictly twos. So when we have to figure out if a Times B is going to be equal to 2222 times 10 hello. And we&#x27;re not gonna have that case because a Times B A. B is going to be equal to the rows of a multi by the columns of B. And that&#x27;s going to be zero everywhere except for the first position when we have one times one. So doing this will have one times one, which is one. And then we&#x27;ll have zeros twice and then for the second room will have 10 And when we square to those a B square by the rows of the columns again so we&#x27;ll have one times one or one and then one times double zeros or zero. And then the second row will be the same thing 10 So a B squared is 1010 and a square beast word is minimum going to have a to right here in the first row. First position, so those are not equal, and that is enough to prove to us that a B squared. That&#x27;s right. This formally will do it in red. Awesome. Since this is the answer, this is enough to prove to us that a B squared is not equal to a squared B squared

David Barker · Answer

Hey, guys. Interesting one here. Ah, let&#x27;s say that a equals a matrix. Okay, A equals a matrix. Um, two is a two by two matrices Think also call them is a square matrix matrices because it to buy to say my same And b is the matrix negative for eight. One and zero. Now, here&#x27;s I&#x27;m question there&#x27;s a be here equal the hey, I Was this times this the same as this Times this on generally you think? Yes. Because, um, what is it? The community of property of multiplication that usually can multiply things in whatever order you want. Um, but we&#x27;re gonna see it&#x27;s not true with matrices. All right, so let&#x27;s just find out when you do a times B first. All right, so that looks like let&#x27;s get rid of they here. I raised, uh, So here&#x27;s a Okay, put it down here and be waas. We&#x27;re multiplying it. Negative for eight. Uh, warm and zero. Okay, this would be a times B. What is this April again? We&#x27;re gonna take the rose rather times this call them and corresponded to a negative full. That would be negative. A And then the second with second negative one times a Combine it with a negative A. Okay, Okay. Okay. Next, we&#x27;re gonna multiply that row with this column. That&#x27;s gonna give us the number there. Two and one is two. Combine it with negative one and zero will only think time. 00 Okay, now we&#x27;re gonna do the same thing with this row. First of this column. Three and negative full. That&#x27;s they give 12. Combine it with five and eight. That&#x27;s full tea. And last three and 13 times one is three. Combine it with five and zero. That zero again. All right, Now let&#x27;s combine. These will do on the second page. Okay, So negative. And a negative eight with same sign. This is a times B by the West upset. So Nao, they and a negative eight combined is a negative 16. Okay, then we have two and zero on three and zero. What can remember that? That&#x27;s two in three. Right? Is to me no. Two and three. There we go. And what was this one negative Before we were these signs, A different summits attract what&#x27;s 40. Take away 12. That&#x27;s 28. And the bigger number here is the positive fully. So my answer is positive. So I&#x27;m done. That&#x27;s eight times a day now. What&#x27;s Billy Times? A. Is it the same? All right, get rid of that. Like that. So let&#x27;s do it again. But this time I have to switch these olders around. Okay, Mr Raisa. So this time I&#x27;m gonna do bay fists. Negative. Full a warm and zero times a 23 Any guesses? What, you think I could have given away a little bit? All right, let&#x27;s go ahead and see what we get. This row times this column, Not your four times too negative A. I&#x27;m gonna combine it with one and three. What&#x27;s one times three? That&#x27;s three. Okay, now, But multiply these again with bomb called upside right now. Get four times. Was it false? I&#x27;m right by a window. You hear the motorbike? Goodbye. All right. And one on five. That&#x27;s a fine. Okay, on. Now this road to these columns 8 to 16. Zero and 3081 negative. Age zero and 50 Okay, here&#x27;s bay. A no smoke your mumps. That negative. And three. That gives me negative five right for 15 that gives me nine on and he&#x27;s going to say the same 16 and negative A. So are these the same one? No. So therefore get right fancy here. Therefore I be does not equal the A.

Problem

Verify the statements in Exercises $19-24 .$ The …

Problem 20 Easy Difficulty

Verify the statements in Exercises $19-24 .$ The matrices are square.
If $A$ is similar to $B,$ then $A^{2}$ is similar to $B^{2}$

Answer

$A^{2}$ is similar to $B^{2}$

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

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$A^{2}$ is similar to $B^{2}$

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

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

Linear Algebra and Its Applications