verify-the-statements-in-exercises-19-24-the-matrices-are-square-if-b-is-similar-to-a-and-c-is-simil

Question

Verify the statements in Exercises $19-24 .$ The matrices are square.
If $B$ is similar to $A$ and $C$ is similar to $A,$ then $B$ is similar to $C .$

Chris Trentman · Accepted Answer

whereas to prove that if the Matrix B A similar to matrix a matrix see a similar to matrix A than be a similar to seat well, because A is similar to be and is similar to see, this means that there exist in vertebral matrices, P and cue such that p in verse. P B p is equal to a and que in verse. See Q equals a. Then it follows that since both of these air equal to a P in verse, BP is equal to queue in verse. Si que so now let multiply by Q and right multiplied by Q In verse, we obtain que p in verse be p que inverse is equal to see since acute hymns Cuban versus the Identity. So it follows that C is equal to Q p Inverse B P Q. In verse, which we notice is the same as mhm p que in verse in verse be p que inverse And here, um, adding the parentheses for emphasis. So we have that the convertible matrix P Q. Inverse is such that Cuco inverse inverse times B times P Q. In verse equals C, and therefore we have. That deed is similar to see

Runpeng Li · 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.

Tiffany Tran · Answer

So we want to know if this equation is true. So you want to multiply the Matrix to Nick 135 Make sure stay times the quantity of be time. See before home. Eight. Soon. What a multiply. Where to me. So you want to see if that is the same as the other side of the equation? So you want to multiply? This&#x27;ll part first. Okay, so this is gonna be another two by two matrix when you multiply it out will be negative. Four times three will be negative. 12 and then negative for I mean And then one times negative one will be negative. One. So negative 12 plus Make it. If one would be making a 13 and you multiply with the second column. Negative. Four times twos. Negative eight. And then plus one times two would be chosen at those together Will be negative. Six. They&#x27;re going to do the second row. Eight times three is 24 0 time thinking of one will be zero. So this will be 24 and then negative eight times two of 16 plus zero times to be 16. I mean, zero times 20 and then When you add together, we 16 and then you&#x27;re gonna multiply this matrix wave matrix A. I&#x27;m gonna dio this road times this calm again. It will also be a two by two matrix. So two tons negative 13 is negative. 26 negative one times 24 7 Dated 24. When you add those two numbers together, it will be negative. 50. My gonna multiply it with next column two times negative six. It&#x27;s negative. 12 Negative one times 16 will be negative. 16. When you have those two, it would be negative. 28. They&#x27;re gonna move with the second part. The second room three times Negative 13 negative 39 5 times 24 is 120. So when you add those two together, it will be 81 three times negative. Six is negative. 18 5 times 16 is 80 when combined. Those to be 62. Now we want to see if this is gonna be equal to the quantity of a bee matrix of a B. So that will be to make it one. That&#x27;s a negative fund. 35 matrix B will be negative for those. That&#x27;s right. Negative for 18 Cyril. So you want to multiply this first and then you&#x27;re gonna multiply it with the matrix C? Sorry. That&#x27;s a three. You try that again? Me to Oh, negative one to All right, So let&#x27;s multiply this first two times in the forest. Negative eight. Negative. One times AIDS is gonna be native. Eight. So you combine those two, it&#x27;s gonna be negative. 16. Then you can multiply with the second roll. Two times. One is two negative. One times zero It&#x27;s Cyril&#x27;s that that would be it too. They&#x27;re gonna multiple the second role. Three times. Negative force Negative. 12 5 times eight is 40. When you combine those two, it&#x27;s gonna be 20 aids. Three tons. One is 35 times zero is Cyril. So it&#x27;s gonna be three and you want a multiply it. But the next matrix So again, it&#x27;s gonna be this role multiplied by this first column and it will be a two by two Negus 16 times three is negative. 48 2 times negative. One is thinking too. So you combine those two. It&#x27;s gonna be negative. 50 being a multiply with the second column Negative 16 times two is negative. 32 2 times two is gonna before. When you combine, that will be negative. 28 28 times three is 84 3/10. NATO one is negative. Three. When you combine that it would be 81. And then you got multiple with the second column 28 times to its 56 3 times to six. When you committed that, it will be 62 little 62. And then now we can see that these two laters are equal.

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.

Tiffany Tran · Answer

So they want to know if this equation is true. In this case, C is three unit, three times The Matrix is of A and B, which is to over three. And multiply that with the matrix B. Okay, so in this case, you&#x27;re going to multiply this role with this column. So one times negative five is thinking of five negative two times four is negative. Eight. When you combine that, you will be negative. 13 I think in a multiple with a second role. Negative. Positive one. Sorry. Positive. One times choose to make it a two times three is gonna be native. Six. When you combine those two, it will be negative, for they&#x27;re gonna multiplies the bottom row. Four time thinking of five negative 23 times four is 12. When you come by, that is negative eight. And then you&#x27;re gonna multiply with the second column four tons to its 83 times three is nine, which will make a total off 17. And then what you want to do is take this three and you&#x27;re gonna multiply it to each number. So three times negative 13 will be negative. 39 three times negative four will be negative. 12. The re technical of eight is negative. Tony, for this school in distributive property Redistricting. The three to all the numbers in the Matrix three times 17 is 51. Now we want to see if this answer will be the same. Us on the right size of love. Matrix of a is one negative. Two for three multiply. In this case, remember, three C is your three. So three times the matrix of B this the matrix that be again, you&#x27;re going to use a distributive property. So you want to multiply this three to all the numbers inside. So three times negative five is gonna be negative. 15 three times two is 63 times four is 12 and then three times three is nine. Then we&#x27;re gonna bring down this matrix down here, 14 to 3. Again, we are going to multiply Matrix ease. So this road times this column So one times negative 15 is negative. 15 2 times 12 is negative. Tony. Four win Combined. Those two, you&#x27;re gonna make negative 39 one time six. I&#x27;m sorry to do this. Calm here. So one time. 66 negative two times nine is negative 18 which will make negative 12. And then you&#x27;re gonna do the second roll four times. Sinking a 15 is negative. 63 times to all this 36. When you combine, those two is thinking to 24 four times six is 24 3 times nine is 27. When you do 24 plus 27 you will also get 51. So in this case, these two natures.

Problem

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

Problem 21 Easy Difficulty

Verify the statements in Exercises $19-24 .$ The matrices are square.
If $B$ is similar to $A$ and $C$ is similar to $A,$ then $B$ is similar to $C .$

Answer

The above relation shows that $B$ is similar to $C$ .
Thus, the original statement is verified.

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

Linear Algebra and Its Applications

Kayleah T.

Kristen K.

Samuel H.

Joseph L.

Vectors Intro

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The above relation shows that $B$ is similar to $C$ .Thus, the original statement is verified.

Linear Algebra and Its Applications

Kayleah T.

Kristen K.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Kayleah T.

Kristen K.

Samuel H.

Joseph L.

The above relation shows that $B$ is similar to $C$ .
Thus, the original statement is verified.

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

Linear Algebra and Its Applications