🔷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....
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[@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

[@Sylviadaniel]

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God bless you ❤

[@SkanCityAcademy_SirJohn]

Amen. And good luck

[@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?

[@bobrobert8684]

Excellent tutorials. Thank you.

[@SkanCityAcademy_SirJohn]

You are most welcome, and good luck

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[@SkanCityAcademy_SirJohn]

Thanks so much for your comment and encouragement.

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[@SkanCityAcademy_SirJohn]

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

[@kwabenablessed4888]

Very clear explanations. This was very helpful. Thank you

[@SkanCityAcademy_SirJohn]

You are welcome

[@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
th-cam.com/play/PLInywrvFyvq7oAlPscVnXsd8CRTsh0b77.html

[@D17D]

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

[@SkanCityAcademy_SirJohn]

Thanks so much for watching, 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

[@SonnyTechAcademy]

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

[@SkanCityAcademy_SirJohn]

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[@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.

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

[@SkanCityAcademy_SirJohn]

That's great. Best wishes

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What a good teacher so precise

[@SkanCityAcademy_SirJohn]

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Best teacher ever you are GOAT.

[@SkanCityAcademy_SirJohn]

Aww... Thanks so much. Keep watching for more

[@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

[@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

[@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❤️❤️

[@yahyadiaa9679]

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

[@SkanCityAcademy_SirJohn]

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[@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

[@ai_enthusiast78]

amazing teaching method

[@SkanCityAcademy_SirJohn]

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[@chaimaahidji]

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[@SkanCityAcademy_SirJohn]

You are most welcome

[@scenicsceneBD]

It’s to much helpful, love you man ❤❤

[@SkanCityAcademy_SirJohn]

Thank you so so much

[@nehemiahbalozi5731]

Well understood... Thanks

[@SkanCityAcademy_SirJohn]

Most welcome

[@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?

[@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

[@selly0072]

God richly bless you🙏🏽

[@SkanCityAcademy_SirJohn]

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[@cxndy_mocha8076]

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

[@SkanCityAcademy_SirJohn]

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[@edsonsimbaya1993]

Thanks, this is very simple explanation

[@SkanCityAcademy_SirJohn]

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[@SABRINAHAMID-ok3cz]

THANKS A LOT

[@SkanCityAcademy_SirJohn]

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[@SkanCityAcademy_SirJohn]

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[@wannurfatimahayunibintiwis2844]

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[@SkanCityAcademy_SirJohn]

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[@SkanCityAcademy_SirJohn]

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[@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

[@ut_9882]

sorry sir i think there is a small mistake in the value of λ=1,λ=2 and it is equal to λ=-23 not -21

[@SkanCityAcademy_SirJohn]

Please double check your answer, the right values are 1, 2 and 21.

[@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.

[@OdongoKizito]

Thank for the wonderful explaination

[@SkanCityAcademy_SirJohn]

Most welcome

[@pankajthakur8663]

Excellent

[@SkanCityAcademy_SirJohn]

thanks so much

[@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 😅

[@helifonseka9611]

Thank you from Sri lanka! 🙏

[@SkanCityAcademy_SirJohn]

Youre most welcome

[@efosaomoregie5246]

Thank you bro we love and appreciate you

[@SkanCityAcademy_SirJohn]

You're welcome

[@SkanCityAcademy_SirJohn]

Thanks for watching

[@Twilightaria]

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[@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?

[@MORINGELOMANYAKI]

Nice and reasonable solution

[@SkanCityAcademy_SirJohn]

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I think you need an oscar award🥳🥳🎉

[@SkanCityAcademy_SirJohn]

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its so tebeda thanku

[@SkanCityAcademy_SirJohn]

You are most welcom

[@cherrybuff5991]

Thank you from India♥

[@SkanCityAcademy_SirJohn]

You are welcome

[@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.

[@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

[@everything4editing.]

Thanks so much ❤❤❤

[@SkanCityAcademy_SirJohn]

You are most welcome

[@ssalijovan123]

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

[@SkanCityAcademy_SirJohn]

Amen. No please

[@reatbang]

𝐓𝐡𝐚𝐧𝐤 you

[@SkanCityAcademy_SirJohn]

You are most welcome

[@samaawagih7272]

Spectacular Explanation.

[@SkanCityAcademy_SirJohn]

thanks so much Wagih

[@OsazuwaEro]

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

[@SkanCityAcademy_SirJohn]

Smooth draw

[@manuelmakritos]

Wow .....I love this explanation

[@SkanCityAcademy_SirJohn]

Thanks so so much

[@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

[@georgeasabre6393]

You be doing the most 💪🤲

[@SkanCityAcademy_SirJohn]

thanks so so much. good luck today

[@user-ru4vf5se2s]

Thank you very much

[@SkanCityAcademy_SirJohn]

You are most welcome

[@mcnosike7935]

Thank much for this video it really help

[@SkanCityAcademy_SirJohn]

You're nost welcome

[@InthipornBunkhan]

My textbook said λI - A and your is A-λI. Is these two method have a different answer? Because at the start I use λI-A but the rest I follow your method.

[@SkanCityAcademy_SirJohn]

Well you can solve more questions with that approach to see if the answers will be the same, but then my method is what you see in most textbooks.

[@garpthehero3221]

god bless you thank you so much

[@SkanCityAcademy_SirJohn]

thanks so 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

[@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

[@edvinaleksandrov1417]

very good explanation

[@SkanCityAcademy_SirJohn]

Thanks so much Edvin

[@sajjalsayjal3640]

How we find these eigen values that you write??

[@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

[@meshackmwaniki924]

Wonderful sir.

[@SkanCityAcademy_SirJohn]

Thank you very much

[@Dee_alh]

I wish my professor explains well like you

[@SkanCityAcademy_SirJohn]

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

[@norgac9103]

Excellent explenation. But one point. How i get lamba 1,2,21 without calc ?

[@SkanCityAcademy_SirJohn]

On your calculator, press mode, then equation in the form ax³ + bx²+cx = 0
Then type in the values of a b c and d as in the equations

[@norgac9103]

​@@SkanCityAcademy_SirJohn And if i cant use calc i must use cubic equation or is there another variety ?

[@SkanCityAcademy_SirJohn]

@@norgac9103 you can use the factor theorem

[@norgac9103]

Thank you .

[@norgac9103]

Bro can I send you one example on custom vectors. I've been counting for maybe 3 hours and I can't get to the vector. I'll send you some money for coffee if you want :D

[@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

[@curtixscapparrotti8141]

well simplified. Gracias

[@SkanCityAcademy_SirJohn]

Thanks so so much

[@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

[@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

[@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 ?

[@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

[@henokbezabih8648]

Thank you very much Sri

[@SkanCityAcademy_SirJohn]

You are most welcome

[@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

[@ghost3826]

Goated 🐐

[@SkanCityAcademy_SirJohn]

Thank you very much

[@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

[@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

[@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.

[@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

[@nyambemukena5318]

How did you jump to 21 as the value lambda

[@SkanCityAcademy_SirJohn]

Those are the roots of the equation

[@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.

[@NeverTHOUGHTofIT]

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

[@SkanCityAcademy_SirJohn]

Okay...noted

[@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

[@bestanimerecaporiginal]

Please man what software do you use

[@rizwann098]

Love from Kashmir 🍁❤️

[@SkanCityAcademy_SirJohn]

Thanks so so much

[@rizwann098]

@@SkanCityAcademy_SirJohn it's my pleasure to get a teacher like u ... I'm pursuing masters degree in economics but maths teacher isn't so good that's I was finding a teacher who can explain these things straight forward....
❤️❤️Thank u so much again sir

[@SkanCityAcademy_SirJohn]

@rizwann098 you are most welcome

[@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).

[@reginaldgoka5224]

Coming in clutch I see

[@SkanCityAcademy_SirJohn]

nice

[@Gaayathri_Ganesh]

Thank you so much!!

[@SkanCityAcademy_SirJohn]

You are welcome

[@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

[@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.

[@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

[@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

[@pianielmdk]

God please help me remember all this for my exam 🙏🏽😢

[@SkanCityAcademy_SirJohn]

Amen.
The LORD is our helper.

[@pianielmdk]

@@SkanCityAcademy_SirJohn thanks 🙏🏽 the question on eigenvalues contained 25% of the marks

[@Algorithmic26]

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

[@Geeta22.08]

🎉 thankyou

[@SkanCityAcademy_SirJohn]

You are most welcome

[@karidjatoucisse3212]

great jobbbbbb. thanks

[@SkanCityAcademy_SirJohn]

You are welcome

[@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?

[@prosperchidexogbodo5386]

Instead of doing that just factor your rh values