without-using-row-reduction-find-the-inverse-of-aleftbeginarrayccccc1-2-0-0-0-3-5-0-0-0-0-0-2-0-0-0-

Question

Without using row reduction, find the inverse of
$A=\left[\begin{array}{ccccc}{1} & {2} & {0} & {0} & {0} \ {3} & {5} & {0} & {0} & {0} \ {0} & {0} & {2} & {0} & {0} \ {0} & {0} & {0} & {7} & {8} \ {0} & {0} & {0} & {5} & {6}\end{array}ight]$

Jack Chen · Accepted Answer

you&#x27;re so for this program, we can petition the metrics A into a Brock metrics. So, um, ever use? We can petition the metrics in to be zero zero zero c zero 00 De so after this part petition, we have a magnum broked magics A where ve caused to 1235 That means beaver Sequels to minus 5 to 3 minus one. See those two? Ah, one by one matrix to so see humorously closed toe. Just 1/2 the d equals two a two by two matrix 7856 That means at the mercy close to three. Minus four. Minus five. Over to seven. Over two. Since on this universe, you most mattresses are really easy to compute. Um, so we can find this result and we directly have a university course to ewers seen universe and the b humors, and we just put the rest entrance by zero. So we have minus five to 000 three minus 1000 00 1/2 00 on 0003 minus fall and 000 minus five. Half on the 7/2. So this is the universe off A. We values

William Scherer · Answer

number 435. They want us to find the inverse of this matrix here on the left. Okay. And the way we&#x27;re going to do that is to write the identity matrix here on the right and do elementary row operations until we get the identity matrix here on the left. Okay, so this is what we were given. This is a six by six identity matrix, okay? And we almost have the identity matrix here. But we have to get rid off these five ones and row six, okay? And we can do that by subtracting each of these rows from row six. Okay, so we don&#x27;t really need to keep rewriting, um, the whole thing, because we&#x27;re only gonna be working with row six. So I&#x27;m gonna do Row six minus row five, and then I&#x27;m gonna do rose six minus row four, rose six minus row three rose six minus road to and rose six minus roll one. Okay, um, so let&#x27;s start with grow six minus row five. Okay? Everything will stay the same. Except for this right here will be one minus one. I&#x27;ll get a zero right here. So let me let me just cross this out for now and rewrite it as a zero. And also right here. I&#x27;m gonna end up doing zero minus one. So I&#x27;m going across this out and write negative one, Okay? Nothing else changes because you&#x27;re just subtracting zeros, except for right here and right here. Okay, then let&#x27;s subtract Rose six from row four. Well, everything will stay the same. Except for one minus one. Right here. Somebody across this out and right zero. And also zero minus one right here will give me a negative one. Someone across this cell and write negative one. Okay. And it keeps going like that. Rose six minus row three. The only one that changes is right here. When I do one minus one, this becomes zero. And when I do zero minus this one, This becomes negative one. Okay. Rose six minus wrote to one minus one. I get a zero right here and zero minus one. I get a negative one right here. And finally Row six, minus row one one minus one. I get a zero right here and zero minus one. I get a negative one right here. So now we have the identity matrix on the left. We have all zeros. Except for the main diagonals, all ones. So this matrix here is our inverse matrix. Okay, so, um, just to clean this up, everything on the bottom, everything on the bottom row is a negative one, except for the bottom. Right. So these are all negative ones on the bottom row, except for the bottom. Right. And this matrix here is, let&#x27;s say the original matrix was a So this is the inverse matrix. A eight of the negative one. Okay. And of course, this matrix on the right. I mean, on the left is the identity matrix. So this one here on the right is the answer.

Mauricio Araiza Canizales · Answer

So we&#x27;re given the following matrix, which we may call eight a matrix and want to find a embers. So the purpose of this video is to find a members and to find eight members were gonna document our matrix eight. Just gonna be 1 to 0 to three sirrah to one Sierra zero sirrah, Sirrah. Three, sir. One syrup to zero. Sierra one. Sarah Sarah. 12 0 Sorry. 101 to Sarah with the identity matrix. So in this case, the identity matrix free three. Basic sorry for a five by five, it&#x27;s simply one 001 001 zero Sarah 01 000 01 And so we&#x27;re gonna use row operations so we can soul for it The Matrix, the inverse matrix. And in doing so, we want to transform a matrix into our identity matrix. And in doing so, our identity matrix will become our eight chambers. So let&#x27;s start off. So the first step, we&#x27;ll be to the buy art too right to to find a new torture. So we&#x27;ll have one, two zero, two and three for first row. Then we&#x27;ll have zero one 1/2 zero and zero then zero 03 this year. One first third brew zero 2001 and finally will have 001 true zero On the rent and sail, we have 10 Sirrah, Sirrah, Sirrah, Syrah 1/2 And the rest is just serious. Then we&#x27;ll have to Syria&#x27;s in a one and the rest of her matrix on the right hand side. Sorry. Our in the right hands say stays the same. No. The second step Well in bulk to multiply our two by native to and added to our one. Then we&#x27;re gonna also want to play our two by negative too. And add it to our four. So the purpose of this second step is to get to eliminate all of the numbers in the second column. So they become zero, except for the only one that is located in Wrote to and its second position. So the first row becomes 10 Negative. One two and three. Second row is 011 Half 00 Their row 003 Sierra one fourth row 00 Native 101 fifth rose zero Sierra one two and zero. There right hand site becomes one degree one. And the rest is you&#x27;re serious. Second roast is the same. Third room is zero 01 Never seen zeros. Fourth row is zero native negative one zero. Oh, sorry. 010 Finally, our fifth bro stays the same with zero at the end. For our 3rd 2 step, we&#x27;re gonna switch the negative. The native of our four with a row three sort of leaking have a one in the third position in the third item in row three. So the first and second roasting the scene, I&#x27;m just gonna rewrite them. So over here, you can notice that this matrix, it&#x27;s giving us a lot more to work with than a two way two or 333 So the thorough become 00 10 and negative one. And on the right hand side will have 010 negative one and zero or 1/4 rope. He&#x27;s now 003 01 with 000 of sarah 01 00 and her fifth rose to the same. The next rule operation. It&#x27;s gonna be multiple off them, so step for we&#x27;ll have multiple row operations. So have I at row three to row one to find a neuro one. We&#x27;re also need to want to play our three by native 1/2 an hour to wrote to. We also need to multiply row three by negative three and added to row for And finally, we&#x27;re gonna need to multiply a negative or three and added to our 5 20 Nero five. And remember that we&#x27;re doing all of this row operations so we can have all of these terms right here and over here to be zero. So performing the separations, we&#x27;ll get the following column one zero Ciro Syrians era than zero. 10 00 zero Sierra one and then two zeros. Our third, Cullen becomes too zero zero. I&#x27;m just gonna place this this here over here so that they are a line and you can see it better. So I 4th 0 and then two, and our final column is just too 1/2 negative. One four in one. On the right hand side, the first column is 100 negative 10 are a second column. Sorry, our second row. Jessie&#x27;s is just 1/2 on the first on the fourth item. Sorry, Arthur Brose. Just zero 10 91. Sirrah, There rose zero negative. 3130 Fifth row is now zero native, one zero in a one on one. So the fifth step involves dividing. I thought row five by two and switching it to our four with our four. Sorry. And we&#x27;re doing this because we want to place these two over here, which, after we divided by two is gonna be one over here. And it&#x27;s just easier if we just divide and switch the rose over here. So after we do that will obtain the first row like this. 100 212 Our second rate. Just 01 Ciro Ciro in 1/2. Thorough. 001 zero negative one. After we have divided by two, our fourth through just become 000 one in 1/2 and our fifth row just have zero syrah syrah and four at the end. The right hand side for the 1st 3 rows ist is the same. So I&#x27;m just gonna rewrite them over here, are forced to throw, become zero negative. 1/2 0 1/2 in 1/2 and our fifth row. This is the same our sixth step symbols eliminating Mr right here so we can have all of this column full of Syria&#x27;s. Except for this one right here. So to do so, we&#x27;re gonna need to multiply our four by native to and added toe or no one. So our first row, yes, becomes one 0001 And the risks is the same. Now, on the right hand side will have won one zero negative too, and negative one. And the rest is the same. So I&#x27;m just gonna rewrite that so we can proceed to the following step so you can clearly see that by doing so, we&#x27;re turning a into I You can kind of see it right here. So right now we have almost other Yanal full of ones. We have already made this section full of serious, and we&#x27;re only left to train this section over here also in two zeros. So the seventh step, it&#x27;s gonna complete this they have no full of once. So we&#x27;re gonna be by our five by four to find a new or five and to do so, we&#x27;re And after that, we&#x27;re gonna multiply our five. My native one and add it to our one to find in your row one. We&#x27;re also gonna move to play our five by one negative 1/2 and added to our two are five is gonna be added to our three 20 and two or three native 1/2 or far. Five. It&#x27;s gonna be added to our four to find a new are for and to remember that this seriousness the steps come after we kept month multiply our five by 1/4. So by doing so, we have already completed our matrix by over here. So in this matrix, I is the analyst full of ones. And the rest is just zeros. No, he&#x27;s right hand site, which is gonna be eight members is gonna be 11.75 Negative. One over four Negative. 2.75 Negative. One zero. 0.375 Negative. One over eight won over eight zero 01 Over. Four. One over. Four negative. One over four zero. Then zero Nearly one over eight negative won over eight again positive. One over eight in one pass and finally will have zero negative. Three over four one over four three over four and zero and on a new whiteboard, I&#x27;m gonna write eight members. So this first column is just 100 zero second column was 1.75 0.375 one over four negative one over eight Negative. Three over four negative one over four negative. One over eight, one over four negative, one over eight negative. One over four Negative. 2.75 won over eight. Negative one over four won over eight. Three over four. Finally, we&#x27;ll have one negative 100 1/2 and zero. And this is the solution to this question. This is the inverse matrix, a buffet which was given the beginning.

William Scherer · Answer

number 431. They want us to find the inverse of this four by four matrix here. Okay, left the bracket open. Because the way we&#x27;re going to do this, we write the identity matrix on the right. Okay, so four by four identity matrix on the right and we want to perform role operations until we get the identity matrix on the left. Okay, so we&#x27;re gonna work one column at a time. We already have a one in the upper left and all zeros in the first column. Okay, Second column. We want to get a one here and the rest zeros. So since we want a zero right here, I am going to subtract row two from row three and put that in Rose three. Okay, so we&#x27;re doing Rose three. Mine is wrote to All right, so rewrite the first two rose. No. And each element in row three subtracted from row 20 minus zero is 01 minus 01 minus 101 minus zero is one and so on. Yeah, and just rewrite Row four. All right, So, row, uh, column two is finished as we want it. Okay, Column three. We want one right here, and we want the rest of them zero. So I&#x27;m gonna dio row one minus row three and put it in Row one and row four, minus row three and put it in a row for Okay. So I&#x27;m killing two birds with one stone here because I want to get a zero right here and a zero right here. And when we subtract these ones from this one that will give us the zeros we need. So each element and row one minus each element in row three and that will give us 1001 and 11 11 negative 10 re copy wrote to re copy Row three and each element in Row four minus each element and row three one minus negative. One is going to give us a to 01 negative one and one. Okay, so our third column is complete Now for the fourth column. We want this value here to be a one, and we want the rest of these to be zero. Okay, so I am going to multiply row for times one half, and that will give us a one right here. Okay. And then I am gonna add half. I&#x27;m gonna add that to Row three. So half of our four plus row three. And I&#x27;m gonna subtract half of our four from row two. And I&#x27;m going to subtract half of our four from row one, minus half of our four. Okay, so when we do, half of our four will get a one here. And then we only added to this one will get a zero when we do, one minus half of two will get 01 minus half of two will get zero. So that&#x27;s what we want. So pretty easy, operations. But we&#x27;re doing four rows. All right, so in the first row, first row, let me see. We&#x27;re doing our one. Okay? 10 All zeros said for the top left, and we&#x27;re getting one half negative. One half negative. One half, second row. Um, we&#x27;re getting a zero half. Half negative. One half, Third row, zero negative. One half. If half and forth. Row zero, half negative. One half. Half. Okay. So now we have the identity matrix on the left. See, we have all ones in the main diagonal and old zeros for the rest of the elements. So this this matrix on the right is your inverse matrix. So if the original matrix was a this is how we would write the inverse matrix. Now, in their answer, they factored out a one half. Okay, so let&#x27;s just do that. So a half times what will give us one? Well, I have times two will give us one. And this is one negative one negative one. This is zero one one negative one. This is zero negative. 11 one. And this 01 negative. One one. Okay, so we factored out a half. So wherever there&#x27;s a one half, it&#x27;s just one, because a half times one will give us a half where there&#x27;s negative one half its negative one and so on. Okay, so this matrix with the one half factored out is our inverse major

William Scherer · Answer

number 433. We have, ah, four by four matrix here, and we want to find the inverse. Okay, so the way to do that is to write the Matrix, and then on the right hand side, we right the identity matrix of the same size. So all zeros, except for ones in the main diagonal. And that&#x27;s the identity matrix. And we want to perform row operations until we get the identity matrix on the left. Okay, so starting in the first column, we have a one in the upper left, which is what we want, and we want the rest of the numbers to be zero. So I am going to subtract. Um, row three minus row one. Okay. And I am going to subtract. I&#x27;m going to add row four plus five times, row one. Okay, so when we do row three minus roll, one will get a one minus one will get a zero here. And when we do row for plus five, row one will get negative. Five plus five and we&#x27;ll get a zero. Here is Well, okay, so rewriting the first two rows. Okay. Now we&#x27;re subtracting each element from road um, from Rose, three. I mean, we&#x27;re subtracting each element from World one in row one from row three. So one minus 104 minus negative. Two is six. And so on. Okay, now we&#x27;re adding row four plus five. Row one negative. Five plus five times one is zero zero plus five times negative. Two is negative. 10 and so on. Okay, our first column is complete. Now, on the second column, Um, we want to get a one in this position, which we have already, and we want to get zeros here, here and here. Okay, so I am gonna multiply row two times two and added to this to get zero, uh, and negative six times wrote to and added to this to get 0 10 times road to and added to this to get zero. Okay, so we&#x27;re gonna dio two road to two times. Wrote two plus row one negative. Six row, two, plus row three and 10. Road to plus row for Okay, So first row. Two time. Zero plus one. Two times one plus negative too. And so on. Rewriting the second row. Certain row negative. Six times zero plus zero. Negative six times one plus six and continue in the same fashion. Okay, so now, finally, 10 times wrote two plus row for well, here we just have zero. 10 times, one plus negative. 10 0 and continue in the same fashion. Okay, so now we wanna work on column three. We want a one right here. So I am going to I&#x27;m going to multiply Rose three times, 1/5. Or should I say negative 1/5. Negative. 1/5 times rose. Three. Okay. And just to make it simple, I&#x27;m gonna erase this instead of rewriting everything. Okay, So here will have a one here. We&#x27;ll have a one, Okay. And, um, multiply everything by hopes. What happened? Multiply everything by negative 1/5. So you should get 9/5 here. Oops. Should get nine fits right here. Once. If 6/5. Negative. 1/5 and zero. Okay. So now we have a new rose three, okay. And we want a zero in place of the three and a zero in place of the 16. So I am going to do negative three times rose three plus row one, and I&#x27;m gonna dio negative 16 times rose three plus row four. All right, so let&#x27;s see. Negative. Three times. Zero plus one negative. Three times zero plus zero. Negative. Three times one plus three and just continue on in the same fashion. All right. Rewriting wrote to rewriting Rose three. Okay, negative. 16 time. Zero plus zero zero. Again negative. 16 times. One plus 16. 0 negative. 16 times 9/5 plus 21 and just continue in the same fashion. Okay, so now we finished the third column, and we need to work on the fourth column. We need a one right here. So I am going to multiply row for times Negative. 5/39. Okay, so these three are still zeros, all right? And I&#x27;m just gonna erase the other terms. So are multiplying each of these elements in row four times. Negative. 5/39. Okay, let&#x27;s see. So we end up with one right there, which is what we want. Negative nine. Over. 39. 46/39. Negative. 16/39 and negative. 5/39. Okay, so now gonna erase this. All right? And we want zeros for these three elements right here. So I&#x27;m gonna do row for, um times. Negative 5/9. I&#x27;m sorry. I mean, row four times negative. 9/5. And add that to row three. I&#x27;m gonna do negative, too. Times row for And Ed, that&#x27;s a row two. And I&#x27;m gonna do 7/5 times row for and add that to row one. Okay, so 7/5 time zero plus one is just one 00 Um, 7/5 times, one plus negative. 7/5 0 and continue in the same fashion. Okay, negative. Two times. Row four, plus row two, we have zero. We have one. We have zero negative. Two times. One plus 20 Okay. And just continue on in the same fashion. All right? Negative. 9/5 times wrote four plus road three. Well, we have 001 negative. 9/5 plus one times 9/5 will give us zero negative. 9/5 times. Negative. 9/39. Plus 1/5. It was 8/13 and just continue in the same fashion and just re copy row for Okay, So now we have, um, the identity matrix on the left, which is exactly what we wanted. And we have our inverse matrix on the right. And really, this would be our answer. But in their answer, they factored out, Um, won over 39. So I&#x27;m going to do that real quick. So if you see a 13 in the denominator, you multiply the numerator by three. Okay, cause 39 divided by 13 is three. So this becomes 32 one negative. Seven 18 negative. 53 32 10. And this is 24. His negative. 36 this is 21 nine negative. 9 46. Negative. 16 and negative five. Okay, so basically, um, divide each of these elements by 1/39. So if the denominators already 39 the answer is just the numerator. So it&#x27;s negative. 9. 46. Negative. 16. Okay, the ones that have 13 of the denominator, just multiply the numerator by three and you&#x27;ll get your answer. So this is the final answer right here. Um and that&#x27;s it. That&#x27;s your final answer.

Mauricio Araiza Canizales · Answer

So we&#x27;re given the following four by four metrics and we went to find its chambers. So in order to do so, we&#x27;re gonna argument his matrix right here with the identity matrix offered four by four. So we&#x27;ll have negative one zero two in five. Sirrah, Sirrah. Sirrah, too. 0 to 81. Ciro. Zero negative. Three Syrian one on the right hand side, perhaps 1000 Cyril one, Sirrah, Sirrah, 00100 Syrah Syrah one, which is identity matrix for a four by four, which is denominated. A Sorry I support. And the reason we do these it&#x27;s because we want to turn a which is this matrix right here into I support using row operations. And in doing so, I, which is this one. This matrix right here will become a embers. So the first step will be to multiply are one by negative one. So we&#x27;ll have 10 native to 85 00 syrup too. Zero to negative. 10 zero negative 301 and then on the right hand side will have a negative one 000 And the rest of the columns, sir, and the rest of the roasts remained the same. The next step will be to mold to ab negative are one plus or four. So our oneness is the same. And our two and three are just zeros syrup too to native 10 Because if we add our one negative are one to art. Four will have zero negative. Three, two and six. Now they&#x27;re right hand. Site becomes negative. One Sierra 10 sirrah. 0010 1001 The next step will be to switch Earth. Really worried about you? We are too. So we&#x27;ll have our one stays the same Now are three. Which is this one right here. It&#x27;s gonna be a switch, and we&#x27;re going to multiply it by 1/2. So we&#x27;ll have zero one negative one cast zero now or three if our previous aren&#x27;t too. So it&#x27;s 0002 end zero negative. Three, two and six. The right hand side will have a negative. One 00 1/2 0100 1001 Next, we&#x27;re gonna multiply Arth too. By three. We&#x27;re gonna add it to our four. So in this case, are one and our two and three rooming the same are for it. On the other hand, just become 00 1/2 in six. The right hand side is needed. One 00 1/2 001 Sorry. 0100 101.5 one. Then we&#x27;re gonna multiply our four right to end it to our three. So in this case, our first three Sorry. Our 1st 2nd 4th roast remained the same. So Arthur Room Sorry. Our fourth row. It&#x27;s equal. So have zero Ciro? Nope. Sorry. And we&#x27;re also going to stretch are for with or three on the same step. So these terms are no longer zero. So I&#x27;m just going to resume. So this terms in the fourth row are no longer Shiro since we&#x27;re switching our four in our three. So they become zero 01 In truth, in her fortunes becomes 002 which was her previous are three on this side. We&#x27;ll have 2032 and 0100 No, we have switched this. We&#x27;re gonna get rid of these two terms right here. And to do so, we&#x27;re gonna multiply it our tree or three by 1/2. We&#x27;re gonna add it to art too. And we&#x27;re also multiplying our three by two. And I didn&#x27;t to our one. So this matrix looks like one 00 19 010 and six 001 and truth and then 0002 The coefficients on the right can cite are just three 064 one, 02 one, 2032 and 0100 The final step. The two final steps are dividing our four white too to find a new war for so we&#x27;re gonna have the following Matrix 100 19 0106 00 1/12 and 0001 The right hand side is 3064 10 to 1 and then we&#x27;ll have 2032 And finally 0 1/2 00 The final step will be to multiply our four by negative 19 and added it to our one. We&#x27;re old. Some months playing are for needed four and adding to our two to find a new or two. And finally we&#x27;re multiplying our fort, my native Truth and Artemis two or three to find a new or three, so this will deal. D Matrix 100 zero Sierra 100 0010 0001 And if we perform, all of the operations on the right hand side will have three negative and I know 0.5 six for one negative. Three two in one two negative six three in two and finally a 01 caf 00 So this right here is our members matrix, which we can write us eight members if we pull a negative 1/2 factor from this Matrix will have negative. Six 19. Negative truth Negative eight Negative, too. 6 94 made it too negative for 12th negative. Six negative four and finally zero negative 100 as her final answer to this question.

Problem

[M] For block operations, it may be necessary to …

Problem 25 Medium Difficulty

Without using row reduction, find the inverse of
$A=\left[\begin{array}{ccccc}{1} & {2} & {0} & {0} & {0} \\ {3} & {5} & {0} & {0} & {0} \\ {0} & {0} & {2} & {0} & {0} \\ {0} & {0} & {0} & {7} & {8} \\ {0} & {0} & {0} & {5} & {6}\end{array}\right]$

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

Linear Algebra and Its Applications

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Absolute Value - Example 1

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Linear Algebra and Its Applications

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

Grace H.

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

Linear Algebra and Its Applications