🔷15 - Eigenvalues and Eigenvectors of a 3x3 Matrix

🔷14 - Eigenvalues and Eigenvectors of a 3x3 Matrix
Given that A is a square matrix (nxn),
Ax = kx -------(1), where
A = an nxn matrix (square matrix),
x = eigenvector of A corresponding to k,
k = eigenvalue of A corresponding to x
It is usually asked to find the eigenvalue as well as the eigenvector that satisfy the above equation.
Notice that we are only interested in the solution with x not equal to zero.
from (1), Ax = kx
Ax = kIx ------(2) ,
(A-kI)x = 0 ----(3)
the system will give a non-zero solution if and only if det (A-kI)x = 0 ,
det (A-kI) gives rise to a polynomial called the characteristic polynomial and the equation formed when det (A-kI) = 0 is called the characteristic equation. The solutions to the equation are the eigenvalues....
Visit channel Playlist for more videos on Engineering mathematics, applied electricity and Basic Mechanics.
Kindly support and Subscribe
Thank you

[@evansokosodo2791]

This is so straightforward. What a good teacher! Many thanks.

[@SkanCityAcademy_SirJohn]

Awww thanks so much

[@azizb.tapeing9249]

So amazing teacher explained clearly. Can I request a lecture on complex root and equal root

[@bitmesrassdsddddsa]

Thanks for existing man

[@SkanCityAcademy_SirJohn]

Thanks be to God

[@percyarthur5556]

16:03

[@SkanCityAcademy_SirJohn]

The reason is that since the RHS is zero, dividing through by 10 to obtain 1 does not change the value of x3, so it can be ignored.

[@peridakingani]

Understandable 😊

[@SkanCityAcademy_SirJohn]

@peridakingani thanks so much

[@kwabenablessed4888]

Very clear explanations. This was very helpful. Thank you

[@SkanCityAcademy_SirJohn]

You are welcome

[@Dee_alh]

you are explaining from the bottom of your heart thank you

[@SkanCityAcademy_SirJohn]

Thanks so much for your comment and encouragement.

[@AbbSalehi]

I have two original equations with three unknowns ( X, Y, Z). I've just added one extra equation to make the original equations solvable. What should I call this adding process in mathematics? I just need the correct wording for that. Any help would be appreciated. Thanks

[@tomasito_2021]

Your videos on linear algebra have so far been very helpful. I'd love videos on Diagonalisation of matrices, coordinate transformations and Jordan block decompositions. Thanks!

[@SkanCityAcademy_SirJohn]

Thanks so much.
Kindly check this playlist
ruclips.net/p/PLInywrvFyvq7oAlPscVnXsd8CRTsh0b77

[@D17D]

Thanks for this. You are explaining directly from your heart, with care and love

[@SkanCityAcademy_SirJohn]

Thanks so much for watching, best wishes

[@SonnyTechAcademy]

Thanks man. Well explained....the video is long but it's worth it :)

[@SkanCityAcademy_SirJohn]

Thanks so much

[@raghavyadav6121]

your videos are really helpful for calculus and linear algebra, thank you!!

[@SkanCityAcademy_SirJohn]

You're most welcome. And thanks for your kind words too.

[@danielkadima571]

with another 3x3 matrix I found the characteristic polynomial, I put the equation which was cubic into the calculator. This way is still difficult to find the eigen values unless I am doing this wrong. So I took the same equation and plugged it into Mathway I found that the roots are decimals?

[@mr2seis388]

Hey buddy, I want to thank you for taking on a matrix without 0's because most of these youtube videos i've come across have 0's at the top or bottom and its annoying because the problem i'm tryin to solve is anything but 0's! Thanks!

[@SkanCityAcademy_SirJohn]

You are most welcome, keep watching for more great content. I really appreciate your comments.
Where do you watch me from?

[@MulindwaAbdallahconc-sh4ct]

What a good teacher so precise

[@SkanCityAcademy_SirJohn]

Thanks so so much

[@sajjalsayjal3640]

How we find these eigen values that you write??

[@palmershot2779]

I've got a test today and this is all. I needed

[@SkanCityAcademy_SirJohn]

That's great. Best wishes

[@petrkasanda4511]

Thanks very much for this teaching
Much love ❤ and respect from zambia 🇿🇲🇿🇲🇿🇲

[@SkanCityAcademy_SirJohn]

Thanks so much, Kasanda, I appreciate it.
Kindly text me on [phone] whatsapp

[@Salamanca-joro]

Absolute cinema! i have final exam on Tuesday and you just saved me

[@SkanCityAcademy_SirJohn]

Aww I guess that's a great feeling for you.

[@calvinbasotho8437]

Hi. I need to know how you simplified that cubid equation to find 3 lambda values

[@SkanCityAcademy_SirJohn]

You can combine the factor theorem and the long-division method to obtain the factors of the polynomial. hope you are familiar with the two mentioned above. Especially with the factor theorem, if f(x) is a polynomial of degree more than one and 'a ' is a number, then if f(a) is zero, then (x-a) is a factor of f(x).

[@edsonsimbaya1993]

Thanks, this is very simple explanation

[@SkanCityAcademy_SirJohn]

You are most welcome

[@masked_man7745]

Explanation is very good and clear. Keep it up.

[@SkanCityAcademy_SirJohn]

Thank you so much

[@masked_man7745]

@@SkanCityAcademy_SirJohn one of these questions came during my exams and I was able to attend it thankyou

[@SkanCityAcademy_SirJohn]

Aww you are most welcome.

[@chaimaahidji]

this lesson is very awesome , thanks so much ☺

[@SkanCityAcademy_SirJohn]

You are most welcome

[@annahkerubo6371]

In finding eigen values of 21, why did we use row two as the pivot row for reduction and not row 1

[@user-gv5dd3bi5l]

answer this question

[@brianomarion]

Got lost after 10:32 what should i be searching for to know how to get the values,
Are supposed to test all numbers from 1 to n until we get 3 values that make it 0?

[@SkanCityAcademy_SirJohn]

i think i get you, at that point you can use your calculator to get the three values, or you can investigate from 1 to n, with constant practice you will know the numbers that are likely to fit the equation. Meanwhile you can watch this video ruclips.net/video/4kOrkFOfCtI/видео.html

[@brianomarion]

@@SkanCityAcademy_SirJohn I feel if i am in an exam room and have to test all numbers from 1 to n, I'm stuck on 1 question the whole time if the number is like 25 for example,
Thanks for the link to polynomials though, this looks promising

[@SkanCityAcademy_SirJohn]

Youre welcome

[@pianolessonswithbless879]

You'll test factors of 42 only...Both positive and negative numbers

[@habib97se]

thank you for the video, you helped med a lot.

[@SkanCityAcademy_SirJohn]

You are most welcome

[@ai_enthusiast78]

amazing teaching method

[@SkanCityAcademy_SirJohn]

thanks so much for your comment. And good luck in your academics

[@OdongoKizito]

Thank for the wonderful explaination

[@SkanCityAcademy_SirJohn]

Most welcome

[@mcnosike7935]

Thank much for this video it really help

[@SkanCityAcademy_SirJohn]

You're nost welcome

[@scenicsceneBD]

It’s to much helpful, love you man ❤❤

[@SkanCityAcademy_SirJohn]

Thank you so so much

[@selly0072]

God richly bless you🙏🏽

[@SkanCityAcademy_SirJohn]

Amen... thank you very much. best wishes

[@georgeasabre6393]

You be doing the most 💪🤲

[@SkanCityAcademy_SirJohn]

thanks so so much. good luck today

[@humzaqureshi1391]

FOR THOSE STUCK ON 11:05:
Apply synthetic division to the lambda equation that is given. Divide the polynomial by (x-1). After doing that, you should get the values (zeros) 1, 2, 21. The reason 1 is included is because the synthetic division ending in 0 allows that factor to be included in your solution as an eigonvalue.

[@DevStuf]

how do you know what to divide by?

[@SkanCityAcademy_SirJohn]

@DevourOrGetDevoured please kindly state the time in the video so I help you out.

[@DevStuf]

@@SkanCityAcademy_SirJohn found out why alr

[@gbgfgfc]

YEP

[@samaawagih7272]

Spectacular Explanation.

[@SkanCityAcademy_SirJohn]

thanks so much Wagih

[@bestanimerecaporiginal]

Please man what software do you use

[@nehemiahbalozi5731]

Well understood... Thanks

[@SkanCityAcademy_SirJohn]

Most welcome

[@nelsonanthony9898]

17:33 why do you pick an arbitrary value for x2 but not x1? Will or does it make any difference?

[@SkanCityAcademy_SirJohn]

Oh no, it doesn't make any difference, you can either choose for x1 then you use that to find x2. It depends on your preference.

[@viktordowa]

But if there is a negative it will definitely affect your answer, won’t it?

[@SkanCityAcademy_SirJohn]

@viktordowa please a negative where

[@alexkim7270]

Wow thanks for the clear explanation! Can I understand why when you interchange the rows in matrix, it doesn't change the final result?

[@Spartacus005]

I think it's because the rows are just stand-ins for the equations and the columns for the variables. Therefore, you can put the rows in any order and still be fine because you can solve the equation system in any order. It is once you change the order of the columns that you run into problems and change the finals result.
If you were to swap Row 1 and Row 2, it'd be the same as completing Row 2 before Row 1. This does not have a bearing on the final result, so you're free to do that. If you were to swap Column 1 and Column 2, you would be switching the coefficients of x1 and x2 variables, which changes the whole system of equations. Is this making sense?

[@SkanCityAcademy_SirJohn]

@Spartacus005 thanks so much for your contribution

[@jaskiratkaur7781]

Hi i need to know that for long division method for finding the eigen values. What do we divide the equation with?

[@SkanCityAcademy_SirJohn]

You just need to use the factor theorem. You put x = a into the cubic function, if it's zero, the x-a is a factor of the cubic function. Which means you already have one eigen value. The you divide the cubic function by the new factor x-a to obtain a quadratic function, then you find it's factors and the corresponding x values

[@jaskiratkaur7781]

@@SkanCityAcademy_SirJohn thankyou so much

[@SkanCityAcademy_SirJohn]

@jaskiratkaur7781 you are welcome

[@efosaomoregie5246]

Thank you bro we love and appreciate you

[@SkanCityAcademy_SirJohn]

You're welcome

[@SkanCityAcademy_SirJohn]

Thanks for watching

[@lauren3441]

When solving for lambda 3, column 3 row 3 isn’t it supposed to be -20? 28:40

[@SkanCityAcademy_SirJohn]

No please, it's -10-(-10) = -10+10=0

[@BADURELGADIR-dd2ck]

thank you for your useful lecture.

[@SkanCityAcademy_SirJohn]

thanks so so much....I'm grateful

[@ChidiebubeAli]

For the first eigenvalue, I thought it should not have zero as a value​@@SkanCityAcademy_SirJohn

[@nesaaqlimakhan]

hey , so for the values of eigenvector , our aim should be making R3 to 0 ?

[@SkanCityAcademy_SirJohn]

not necessarily, the aim is to convert the given matrix to an upper triangular matrix with the leading diagonals being 1. however when there is a zero row, ie a row with all zeros, it should be at the buttom.

[@yahyadiaa9679]

You saved me from failing my exam for the 4th time

[@SkanCityAcademy_SirJohn]

Wow, that's great

[@plantmc9319]

Dang 4 times that’s crazy. Fr though this dude has the best explanation

[@saja_22A]

How did you get out? Lamda? Final output ? ​@@SkanCityAcademy_SirJohn

[@manuelmakritos]

Wow .....I love this explanation

[@SkanCityAcademy_SirJohn]

Thanks so so much

[@Gaayathri_Ganesh]

Thank you so much!!

[@SkanCityAcademy_SirJohn]

You are welcome

[@stevenkanguya5087]

THANK YOU VERY MUCH,,, YOU JUST EARNED YOURSELF A SUBCRIBER

[@SkanCityAcademy_SirJohn]

thanks so much Steven

[@viktordowa]

Do you always have to make the last line to have all zeros or if you want you can just calculate without making the last line all zeros

[@SkanCityAcademy_SirJohn]

Not necessarily, but if there appears a zero row, then it should be at the button

[@cxndy_mocha8076]

This is so easy after listening to this. Tysm! 😭

[@SkanCityAcademy_SirJohn]

Thanks so much for your comments and good luck in your studies.

[@darcash1738]

This is awesome! I was wondering, is the best way for this usually the cofactor expansion? Or if we happen to have 1's in our matrix do you think it is more worth it to do Chio's decomposition to make it one dimension lower? I tried the normal 3x3 trick where we add the first two columns on the outside of it to do that but i found this pretty messy

[@SkanCityAcademy_SirJohn]

Wow, really

[@darcash1738]

@@SkanCityAcademy_SirJohn honestly I don’t know I guess it depends. This cofactor expansion would be nicest in the case everything else were zeros up top. And you have to get lucky for chio bc the whole diagonal is already excluded due to the -lambda part. I learned Chios condensation a bit ago and I think it’s so cool, it’s just that I rarely find a chance to use it 😂

[@SkanCityAcademy_SirJohn]

yes actually@@darcash1738

[@SkanCityAcademy_SirJohn]

where do you watch me from? which program do you read and level?@@darcash1738

[@darcash1738]

@@SkanCityAcademy_SirJohn I’m from America, and I’m just taking some intro to linear algebra class. I like learning math on my own sometimes too so I just happened across Chios condensation one day.
I wish we’d learn more cool tricks like that too. Just right now I learned that the characteristic equation for 3x3 is λ^3 -trace(A)*λ^2+Diagonal Minors(A)*λ - |A| = 0. If you have any cool tricks too (determinants, eigenvalues or vectors, etc), please recommend them even if they might be a bit above my current level 😅

[@Zyscha]

For Lambda= 21, my eigenvectors are coming out to be [6,6,1]. Can you please check yours once? I think you can not perform a row operation using a row if you have operated on that same row in the same step.

[@SkanCityAcademy_SirJohn]

Hi Zyscha, kindly check your approach one more time, if you are still not getting what I had, then you let me know, because what I've done in there is the actual thing.
Thanks so much

[@Zyscha]

@@SkanCityAcademy_SirJohn I don’t know I have done it multiple times, I reach the same answer. How do I show you my approach?

[@SkanCityAcademy_SirJohn]

Please are you on WhatsApp?

[@ace09wrld]

there's a shortcut to the eigen values he solved for and it works;
λ^3 - (sum of diagonal of the matrix)λ^2 + (sum of the diagonal of the adjoint of the matrix)λ - (the determinant of the matrix)

[@SkanCityAcademy_SirJohn]

Thanks so much for input❤️❤️

[@NeverTHOUGHTofIT]

Can you do a video about Eugene roots of symmetric matrix that would be good

[@SkanCityAcademy_SirJohn]

Okay...noted

[@KadmielAcquah]

16:53 For lamda 1 ,i think the matrix was not in its row echelon form,if it was can u explain further??

[@SkanCityAcademy_SirJohn]

It is in Row echelon form. For Row echelon form, diagonal entries are 1 and the elements of the upper triangular matrix can be any other value. Unless in a case where the elements in a row are all zeros, then it is adviced to put that row at the button. While for reduced row echelon looks like the identity matrix

[@SABRINAHAMID-ok3cz]

THANKS A LOT

[@SkanCityAcademy_SirJohn]

You're welcome!

[@MORINGELOMANYAKI]

Nice and reasonable solution

[@SkanCityAcademy_SirJohn]

Thanks so much

[@wannurfatimahayunibintiwis2844]

thank you!!!

[@SkanCityAcademy_SirJohn]

You are most welcome

[@FatawYakubu-908]

Please for the cubic equation if u get the values to be decimals, How do we solve it

[@SkanCityAcademy_SirJohn]

Usually you will get whole number values, if you get decimals, kindly check if the cubic equation is right

[@FatawYakubu-908]

@@SkanCityAcademy_SirJohn okay thanks

[@rivieraokapi]

Thank you my friend, you made it a lot more digestible. What a teacher!!

[@SkanCityAcademy_SirJohn]

You are most welcome. Please keep watching for more

[@kubabak4]

I have a 3x3 matrice [57 0 24, 0 50 0, 24 0 43] and all calculators and solutions indicate that the +-+ doesn't apply. I was wondering why could this be i.e. to get the right answer you must solve it with the negative : : (57-x)(50-x)(43-x) -24(50-x)(24). I expected it to be positive. Any idea why ?

[@SkanCityAcademy_SirJohn]

Are you sure you have punch in the calculator the right entries?

[@kubabak4]

@@SkanCityAcademy_SirJohn So the issue was that I ignored the 0s therefore it was +24[(0x0)-(50-x)(24) instead which is non-intuitive.

[@SkanCityAcademy_SirJohn]

Okay

[@curtixscapparrotti8141]

well simplified. Gracias

[@SkanCityAcademy_SirJohn]

Thanks so so much

[@allstar7778]

Any reason why you are not using krammer's rule which is much simpler than using charachteristic polynomial equation ?

[@SkanCityAcademy_SirJohn]

No reason please, you can use crammer's to solve as well.

[@allstar7778]

@@SkanCityAcademy_SirJohn Alright thanks a lot sir for your reply, your video is really helpful. I thought there must be some mathematical reason. Thanks for clearing this. I also prefer charahteristic polynomial, it somehow just clicks in my brain although it is slow process. One quick question, is it necessary to calculate REF as well for computing an eigen vector ? what if we just a put a quadratic equation directly without computing REF ?

[@user-ru4vf5se2s]

Thank you very much

[@SkanCityAcademy_SirJohn]

You are most welcome

[@miracle_winter6118]

I'm confused....so is it the same for all examples or the swapping and multiplication will vary? Like.....how do you know what to do? Is the bottom row always supposed to have all 0s?? I'm confused...😢

[@SkanCityAcademy_SirJohn]

It varies, it depends on the question you are solving. The idea is, if there is a zero row, then it should appear at the bottom.

[@cclemon2531]

when calculating the eigenvectors in the case lamda equals to 1, can i just let the x1 be 1 rather than x2 be 1?

[@SkanCityAcademy_SirJohn]

Yes, you can

[@helifonseka9611]

Thank you from Sri lanka! 🙏

[@SkanCityAcademy_SirJohn]

Youre most welcome

[@Dee_alh]

I wish my professor explains well like you

[@SkanCityAcademy_SirJohn]

awww thanks so much, where do you watch me from?

[@rexomiv7352]

Great job man

[@SkanCityAcademy_SirJohn]

Thanks so much

[@everything4editing.]

Thanks so much ❤❤❤

[@SkanCityAcademy_SirJohn]

You are most welcome

[@shivanikumari680]

Can you tell me how to find eigen value of this equation x^3+25x^2+50x-1000 ????

[@SkanCityAcademy_SirJohn]

The eigenvalues are
-20, -10 and 5.
Use the factor theorem to do so.

[@edvinaleksandrov1417]

very good explanation

[@SkanCityAcademy_SirJohn]

Thanks so much Edvin

[@user-uh4zf9qz3h]

at 10:49 can you make it clear how did you get lamda 1,2 and 3 also i don"t know how to do it on the calculator if you can reply fast cuz i have an final exam the day after tomorrow

[@SkanCityAcademy_SirJohn]

It's a cubic function and hence you need to obtain 3 roots. To do it on the calculator
Mode
5
4
Type the coefficients of the function a, b, c,d, for any one punch equal to for next, you will get the roots

[@mahmoudelmolla3153]

@@SkanCityAcademy_SirJohn thanks man i appreciate it you are the best

[@SkanCityAcademy_SirJohn]

Thanks so much, El Molla

[@user-iy3rq7zg2v]

Think you sif❤❤

[@Twilightaria]

Godddd bless youuu I've been struggling the wholeee day to understand thisss❤❤❤❤❤❤

[@SkanCityAcademy_SirJohn]

That's great, thanks you got sorted at the end. Where do you watch from?

[@Twilightaria]

@@SkanCityAcademy_SirJohn UAE 🇦🇪

[@SkanCityAcademy_SirJohn]

@idontcare7667 that's fine, im from Ghana 🇬🇭.

[@SkanCityAcademy_SirJohn]

@idontcare7667 Muslim or Christian?

[@garpthehero3221]

god bless you thank you so much

[@SkanCityAcademy_SirJohn]

thanks so much

[@reginaldgoka5224]

Coming in clutch I see

[@SkanCityAcademy_SirJohn]

nice

[@ssalijovan123]

Bless you, but so you have any videos about vector spaces and spaning a vector.

[@SkanCityAcademy_SirJohn]

Amen. No please

[@Algorithmic26]

What's mean by eigen value??
Why do this

[@fediMbf99]

Please how you found lambda with third equation like in the video (10:32)

[@SkanCityAcademy_SirJohn]

It's a cubic equation, just punch on your calculator

[@akhileshakhil4390]

how did you get the roots of the equation, I mean how did you get the eigen values.

[@SkanCityAcademy_SirJohn]

I used a calculator.

[@cherrybuff5991]

Thank you from India♥

[@SkanCityAcademy_SirJohn]

You are welcome

[@pankajthakur8663]

Excellent

[@SkanCityAcademy_SirJohn]

thanks so much

[@henokbezabih8648]

Thank you very much Sri

[@SkanCityAcademy_SirJohn]

You are most welcome

[@ramdanhaerullah6907]

Its detailed, i'm helped

[@SkanCityAcademy_SirJohn]

That's great

[@user-nf2jr2nh2r]

would like to teach me an easy method for getting the eigen vectors than eclon because I have failed to understand

[@SkanCityAcademy_SirJohn]

Really. Sorry about that

[@paulowiredu7586]

From your accent, I could spot you're my Ghanaian brother..... Watching your video from the States.
.

[@SkanCityAcademy_SirJohn]

That is correct. I'm a Ghanaian

[@SkanCityAcademy_SirJohn]

That's great, are you doing postgrad studies?

[@wangster331]

Masterpiece. Writing my exam this morning. It sure would save me 😊

[@SkanCityAcademy_SirJohn]

Yes please, good luck

[@wangster331]

@@SkanCityAcademy_SirJohn thanks, I love you

[@SkanCityAcademy_SirJohn]

@wangster331 aww thanks so much
Where do you watch from

[@wangster331]

From Nigeria

[@SkanCityAcademy_SirJohn]

@wangster331 great

[@OpareAddoNanaYaw-tg8ni]

At 28:04 why was (-10-10) equal to 0. If I’m not mistaken it should be 20.
More clarity on this please

[@SkanCityAcademy_SirJohn]

R2-R3. -10-(-10) = -10+10 = 0

[@user-zj7bh1oo5r]

Thank you!🙂

[@SkanCityAcademy_SirJohn]

You're welcome, where do you watch from?

[@user-zj7bh1oo5r]

@@SkanCityAcademy_SirJohn I am from Kenya.

[@SkanCityAcademy_SirJohn]

That's great, thanks for watching

[@karidjatoucisse3212]

great jobbbbbb. thanks

[@SkanCityAcademy_SirJohn]

You are welcome

[@niamh1735]

for lamda=1 I got [1 -1 0] since I let x1=1 therefore x2= -1, is that wrong?

[@SkanCityAcademy_SirJohn]

That's also correct.

[@niamh1735]

@@SkanCityAcademy_SirJohn thank you very much!!

[@zzx1212]

for 18:00 , can I let x1 be 1 instead ？

[@SkanCityAcademy_SirJohn]

Yes, you can

[@nyambemukena5318]

How did you jump to 21 as the value lambda

[@SkanCityAcademy_SirJohn]

Those are the roots of the equation

[@YasabnehAddisu]

its so tebeda thanku

[@SkanCityAcademy_SirJohn]

You are most welcom

[@meshackmwaniki924]

Wonderful sir.

[@SkanCityAcademy_SirJohn]

Thank you very much

[@sanketkumbhar]

How to find eigen values & eigen vector corresponding to smallest eigen value in 3 by 3 matrix

[@sanketkumbhar]

Plz give me thise question answer

[@Enthub47]

Please can you tell me what app you used for this tutorial. The board and pens style in particular. It’s soo smooth 🙂

[@SkanCityAcademy_SirJohn]

I used smoothdraw

[@Enthub47]

@@SkanCityAcademy_SirJohn no wonder it’s smooth ! You do all 🙌🏾🙌🏾

[@SkanCityAcademy_SirJohn]

Thank you

[@watsonnampon2993]

please could you show how to obtain 21 as the eigen values

[@SkanCityAcademy_SirJohn]

You can basically use a calculator, of the factor theorem

[@diyadiyapp9461]

Thanks 😇

[@SkanCityAcademy_SirJohn]

You are welcome

[@OsazuwaEro]

Thank you sir.. Pls what software do you use?

[@SkanCityAcademy_SirJohn]

Smooth draw

[@angelloparody3216]

why do I have to divide the equation by negative 1?

[@SkanCityAcademy_SirJohn]

Nothing really, just to make the coefficient of x³ positive. But you can ignore it and still get the same answers for x(1, 2, 3)

[@JosephOtieno-zu2rm]

I think you need an oscar award🥳🥳🎉

[@SkanCityAcademy_SirJohn]

Thanks so so much

[@user-qj5fv6ss1r]

Please why did you multiply the equation by -1?
Because I don't understand

[@SkanCityAcademy_SirJohn]

its nothing big, we just want to make the coefficient of lumda cube be +1. You know some people are not comfortable working with negatives