in-exercises-33-36-verify-that-det-e-aoperatornamedet-eoperatornamedet-a-where-e-is-the-elementary-2

Question

In Exercises $33-36,$ verify that det $E A=(\operatorname{det} E)(\operatorname{det} A),$ where $E$ is the elementary matrix shown and $A=\left[\begin{array}{cc}{a} & {b} \ {c} & {d}\end{array}ight]$
$$
\left[\begin{array}{ll}{1} & {0} \ {k} & {1}\end{array}ight]
$$

James Chok · Accepted Answer

Okay, so we want to verify That&#x27;s determinant off E a secretive determinant off e Taiba. Determinants off, eh? So first figuring out what e times is we have one times a plus zero times. See? Excuse you, eh? One times B plus zero times the which gives you be and K Time&#x27;s a plus one time see, which gives you a k A plus c and then you have keytones B plus one times D, which gives you K B plus d. So then the determinants off e A. It&#x27;s simply a became be rusty. One isn&#x27;t b a k a pussy, which is K A B plus 80 Marchioness k a b plus B c this term cattles at this time. So you&#x27;re left with a D minus. BC now taking the tenements. All these disability, we get it&#x27;s evidence. Oh, he&#x27;s even to a D morning specie. The Turners off e is equal to one times one giving you one k time or minus by Kate. I&#x27;m zero excuse zero. This is one. So then determinant. Oh, aye. Was by 10 minutes off e years, even to a d on a species, which is the exact same as taking the tone int off, Peter

Sriram Soundarrajan · Answer

different today. We&#x27;re going to solve problem number 33 here to do Tom and is given by one. Okay. Zero lot determinant off is given the uh huh. The see the so he will be equals a plus. Casey B plus Katie seat The determinant off e equals a press case into the minus B plus Katie into city 80 plus k c V minus BC Linus K. D. C. So it it&#x27;s like a d minus b c. Then this is a This is it determinant off equals what return mint off equals a T minus busy So determinant off e equals determinant off into the dominant off.

James Chok · Answer

Okay, so in this question, we want to figure out who want to show that determinants off E a is equal to determinant off e time flight attendants off, eh? So, working on the left hand side. First, we have to first figure out what ee A&#x27;s. This is 10 k on a B C. D. So one times they give you a plus zero times. See, which uses a one times B plus zero times e excuse would be zero times a plus. Kate M. C. Which is this Casey zero times b plus Que Tom ze, which is just Katie todo determinants. Oh, hey is equal to Okay, well, eight times, Katie, So K a B minus. Casey turns me so K B c. So Okay, plans were a D minus, Casey. Okay, now the term it&#x27;s off, eh? Is equal to eight m c minus anytime soon and determined off E is just one times K one times. Can&#x27;t give Luke a minus zero times here. Is it just k and determinants old, eh? 10 by 10 minutes, we&#x27;ll be is just came from a the artist B C, which is the exact same as determined

Sriram Soundarrajan · Answer

great. We&#x27;re going to solve a problem. Number the foot here. The government is a because mhm b c think it takes equals the 011 Seattle So determinant mhm equals zero minus one. The physicals minus one. So then e equals 0110 into a B C B, which will be getting it does seat the mhm the so determinant off. Yeah, equals C B minus 80. What we should be C minus 80 determinant off. Hey, because a T minus b c so determinant off and in the pit dominant off mhm equals minus learn Hindu A T minus B C to be B C minus 80.

James Chok · Answer

So in this question we were verified That determines off a B is equal to determine it off eight times by the tenement off beat. So let&#x27;s work the left inside out first. So a be is it 23 minus 16 months to okay for three months. One month street, Just you. So this is three times four is 12 miles. Six. That&#x27;s six. Three times three is nine minus six and three is 18 which is collected nine negative one time by four minus two times one. So that&#x27;s negative for months. Makes awful plus two. So that&#x27;s negative too. And negative. One by three, which is negative. Three plus six. Excuse you three to determines. Oh, baby, is it into six and three, which is 18 minus negative to pass by 99 which is just 80. So what is this your now the tenements? Oh, aye. Is it 23 times by minus two, which is negative. Six minus six times modest one which is 96. So what? You are and the tenements wouldn&#x27;t really need to Catholic This spot all his tablet four times on minus three which is negative, tall and three times, right? Minus one, which is negative. Three. And this is negative. 12. Sastri is 90 line. And so tone INTs. Oh, aye. Times by determines. Don&#x27;t be using which is there anything as atonement?

Lindsay El · Answer

to prove the determinant of C A is equal to C squared times into turning. Today we&#x27;re just gonna evaluate both sides and check of the equal. So first evaluating the determinant of C A. To turn it, see a so see a is equal to see Time&#x27;s matrix a soldier&#x27;s walk by everything in it by sea sweep See a 11 see a 12 Same for the bottom row. So now we don&#x27;t value the determined of that are when we&#x27;re evaluating the determinant of a two by two matrix. We multiply that these two values we have. See a 11 time see a 22 and subtract those two months hard together. So this is our determinant of C A. And so now let&#x27;s evaluate the second part of this so c squared determinant of it. So that&#x27;s just first we have to define the determinant of a which is okay 11 times a 22 minus a 21 times anyone to. And so then we want the C squared determine of a response by this by C squared. Well, now I have to check if these two values air equal these two things, April and we can rewrite this 1st 1 We did as that. So we can much by these take these to seize multiply them together. So you c squared a 11 times a 22 And here we have to seize against weaken minus C square a 21 times a 12 Now the ceasefire In both terms we can just take that out And this is equal by the distributed. So, you know, you&#x27;ve C squared times this value and that&#x27;s exactly what we have right here because this is C squared times the same value. And so this statement that we started with is true. First problem should I will look at what happens if we even and by n matrix. So to prove that these two things are also the determinative see a So in an unvarying matrix, we&#x27;re gonna have n factorial terms. So the determine of C A A is gonna look something like this chart the Matrix C A is gonna look like see a 11 So everything&#x27;s just gonna be multiplied by C. We&#x27;ll see. Okay. And then something like this. So when we defined when we find the determinant of C A we&#x27;re gonna have and factorial terms that are gonna have in different values in it. So it&#x27;s a permutation of all the columns. Right? So we have you might have see a 11 see a 22 see a 33 all the way to see a net. This may be one of our terms, so that&#x27;s one. And then we&#x27;re gonna have n factorial so we can add another. See a 12 see a one No, 23 All the way to see a something like four We want to have our roads in order Should do And for something like that so that we have 21 that 21 and factorial these total terrorists And so what we could see is that there are gonna be end season here. So if we want to rewrite these weaken right see to the end times a 11 82 all the way to eight and then And same for this term, Teoh, have NC&#x27;s multiplied by all the different values. And then So we&#x27;re gonna have and factorial be this Well so in the end, once we reduce this look over here we&#x27;ll have see the end times. These 21 values added together so you can just take these two see that ends out and some based in various. So what we see is that we&#x27;re going to get something that looks like this. There will be an factorial terms in here multiplied by C then, and when we look at this, we&#x27;ll see that he&#x27;s 21. Terms in here is the same as the determinant. So this would be the determinant of a multiplied by sea to the end. And so, looking at our first equation off at the top, we found in blue here the determinant of C A and re see that that could be rewritten as see to the end times of determined obey. So that statement is true.

Problem

In Exercises $33-36,$ verify that det $E A=(\oper…

Problem 34 Hard Difficulty

In Exercises $33-36,$ verify that det $E A=(\operatorname{det} E)(\operatorname{det} A),$ where $E$ is the elementary matrix shown and $A=\left[\begin{array}{cc}{a} & {b} \\ {c} & {d}\end{array}\right]$
$$
\left[\begin{array}{ll}{1} & {0} \\ {k} & {1}\end{array}\right]
$$

Answer

the property det $( E A ) = ( \operatorname { det } E ) ( \operatorname { det } A )$

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

Linear Algebra and Its Applications

Grace H.

Catherine R.

Alayna H.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

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the property det $( E A ) = ( \operatorname { det } E ) ( \operatorname { det } A )$

Linear Algebra and Its Applications

Grace H.

Catherine R.

Alayna H.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

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

Grace H.

Catherine R.

Alayna H.

Kayleah T.

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

Linear Algebra and Its Applications