Inverse of 3x3 matrix

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  • Опубликовано: 18 июл 2012
  • Inverse of 3x3 matrix example. Visit Mathmeeting.com to see all all video tutorials covering the inverse of a 3x3 matrix.

Комментарии • 990

  • @zonamade3380
    @zonamade3380 7 лет назад +329

    i thought this video was ending like four times until he kept going..

    • @JuliaChang
      @JuliaChang 5 лет назад +7

      i was praying in every second "god pls stahp now" but no.

    • @Minh-fo5fd
      @Minh-fo5fd 4 года назад

      @@JuliaChang wdym

    • @Osmosis-jy6xp
      @Osmosis-jy6xp 3 года назад

      @@Minh-fo5fd he means the video was so long that he was praying to god that it would end soon because the steps are tedious.

    • @Minh-fo5fd
      @Minh-fo5fd 3 года назад

      @@Osmosis-jy6xp ik i was messing with him

    • @Minh-fo5fd
      @Minh-fo5fd 3 года назад

      @@Osmosis-jy6xp or her

  • @lovesecretangels
    @lovesecretangels 5 лет назад +77

    Bro idk how many videos Ive watched trying to explain this to me.... I finally understand because of you, thank you 😭🙌🏻

  • @neechan03
    @neechan03 5 лет назад +188

    "...which a computer could do in less than a second"
    dont hurt me like this man

    • @sibahlesammkhize1086
      @sibahlesammkhize1086 4 года назад +4

      What app solves inverses,I'm not about to do this entire process for my online exams 😎

  • @robbybobby6464
    @robbybobby6464 10 лет назад +661

    thats a lot of work for a freaking inverse

    • @MadOrange644
      @MadOrange644 10 лет назад +30

      It's simpler than that, but this video is fucking confusing.

    • @hstera
      @hstera 9 лет назад +7

      ***** lol, i agree. This method isn't that hard, and can be done relatively easy with the mentioned steps and some training... That being said, computers are extremely efficient at solving these kind of problems, so there is no point in learning the mechanics of solving them if they're larger than 3x3

    • @arvinroidoatienza7082
      @arvinroidoatienza7082 6 лет назад +3

      Actually, in my case its much easier. in our case, we used to do rref in finding the inverse. and its damn long, especially for 3x3 s.

    • @gytisdramblewolfskis8521
      @gytisdramblewolfskis8521 5 лет назад +3

      this is quite easy and not complicated. But there's a decent amount of shit to remember.

    • @oida10000
      @oida10000 5 лет назад +7

      There are two other methods: Gauss-Jordan elimination and Montante's method check them out here (they are there along with the video method):
      matrixcalc.org/en/#%7B%7B1,2,3%7D,%7B0,1,5%7D,%7B5,6,0%7D%7D%5E%28-1%29
      I watched the video because In was told that the method would be quiet fast for triangular matrices with many zeros (the nice thing about triangular matrices is that their determinate is simply the product of their diagonal (and you can transform a matrix into an equivalent triangular matrix using Gauss)) in the none zero part (like:
      (1 0 0 2)
      (0 1 0 0)
      (0 0 1 7)
      (0 0 0 1)
      but I don't see it yet.)

  • @julian7719
    @julian7719 9 лет назад +60

    Thank you, sir. You have saved me at least 3 hours of studying.

  • @terminatortscc1
    @terminatortscc1 8 лет назад +732

    This is exactly how teacher's are supposed to teach, obviously they don't otherwise i won't be here.

    • @christellemathieu179
      @christellemathieu179 6 лет назад +1

      Bestivitiness right

    • @lilrosiecheeks2143
      @lilrosiecheeks2143 6 лет назад +2

      Tbh

    • @smin4089
      @smin4089 6 лет назад +14

      When the govt takes control of anything, it goes to shit because the govt doesn't need to respond to market/human demands that are constantly changing.
      The private companies/mom&pop/free market is built on responding to the needs of the people/customers.

    • @sajjankumar2399
      @sajjankumar2399 5 лет назад

      ruclips.net/video/t7VHbC2_fTI/видео.html

    • @sajjankumar2399
      @sajjankumar2399 5 лет назад

      jungle305 ruclips.net/video/t7VHbC2_fTI/видео.html

  • @tstone9151
    @tstone9151 7 лет назад +36

    Holy Crap Dude. Honestly thank you so much. My preCalc teacher explained it in a totally complicated way. You made it so simple for me

  • @rsilvas123
    @rsilvas123 9 лет назад +29

    Great video. Thank you for the step by step lesson. This is exactly what I needed.

  • @frymybaconpl0x
    @frymybaconpl0x 9 лет назад +258

    .......im colorblind

    • @philw5699
      @philw5699 9 лет назад +13

      Kevin Bacon I think they have glasses for that now.

    • @wesleyrichardson4940
      @wesleyrichardson4940 8 лет назад +11

      +Kevin “XDG 2014” Bacon Theres a way to get rid of that of lessen the effects. Its called rubbbing shit in your eye, very effective and not costly at all.

    • @moarts3733
      @moarts3733 6 лет назад +25

      go to your eye settings

    • @garveziukas
      @garveziukas 6 лет назад +8

      You should turn off the color blind mode in your display settings

    • @Muhammad-bf2dk
      @Muhammad-bf2dk 5 лет назад +2

      Did you put it in rice

  • @jeffreystevens8042
    @jeffreystevens8042 8 лет назад +71

    way to make me have an existential crisis at the end dude

    • @Matt_Lifts
      @Matt_Lifts 7 лет назад +3

      Yeah the ending made me laugh hahaha

  • @saritar8819
    @saritar8819 10 лет назад +17

    Absolutely love this! You simplified it! And I love the sarcastic note at the end ;)

  • @BooPeeBoop
    @BooPeeBoop 9 лет назад +122

    It's like 3 hours before my exam, and you saved my life! I was trying to understand the A^-1 for so long!! but this is so much better:) #Subscribed

    • @musicstore9964
      @musicstore9964 6 лет назад

      good student 👨‍🎓

    • @arturopena3765
      @arturopena3765 4 года назад

      i agree. never undertood it until now. thanks.

    • @addictedelectro9848
      @addictedelectro9848 2 года назад

      haha its also 2 hours for my exam

    • @cherryx2113
      @cherryx2113 Год назад +2

      I'm literally taking my exam right now (It's open book)

    • @theboldguy.
      @theboldguy. Год назад

      this is far more faster and effiecent way of finding inverse of a matrix .Just imagine how much time you save on exams .

  • @tactic1313
    @tactic1313 4 года назад +15

    I dont get the part where you just multiply with "+" and "-" to the solved det from N
    is it always starting with a plus and then just + - + - + - ? also for a 4by4 matrix?

    • @woodchuk1
      @woodchuk1 2 года назад

      + - + -
      - + - +
      + - + -
      - + - +
      That’s the configuration for a 4x4. It alternates signs between entries in the row, but the initial entries in the rows also alternate signs going down the columns.

    • @tactic1313
      @tactic1313 2 года назад

      @@woodchuk1 thanks ! I managed to pass my exam but this will surely help others in the future ! 😁

  • @C1001t
    @C1001t 8 лет назад +4

    Thank you very much. My lecturer explained it in a more complex way. This makes it so much easier. Finally have hope of passing this module: D

  • @angeliquaserenity5009
    @angeliquaserenity5009 4 года назад +2

    Nice methodology! Your system of multiplying inverse matrices is much more simple than what I was originally taught. Thanks!!!

  • @erickmaverick3128
    @erickmaverick3128 10 лет назад +1

    Thank you so much for making this video! I had such a hard time grasping this concept during summer school. You make a good tutor!

  • @GDInspirations
    @GDInspirations 8 лет назад +114

    This makes me give up on life ! That congratulations at the end ....

    • @shgofftabibi7980
      @shgofftabibi7980 7 лет назад +4

      See the other video called A-1 in 60 seconds so easy

  • @yaraa.eldosoky2283
    @yaraa.eldosoky2283 7 лет назад +3

    This method is much easier than the ones I knew in the past and fastest ....... so thanks a lot👌🖒🖒

  • @danielle_le
    @danielle_le 3 года назад +1

    This was a thorough and well-paced explanation. Thank you so much and keep up the great work.

  • @ari_says
    @ari_says 8 лет назад +1

    This was EXTREMELY helpful! My professor only went over 2x2 matrices, but our study guide includes 3x3 matrices also. You have saved my exam tomorrow!

  • @balasankarajith2923
    @balasankarajith2923 4 года назад +4

    Me at the begining of the video - Ah this gonna be easy, just a short simple steps.
    The steps - *John Cena's voice* : Are you sure about that?

  • @Kaamz1926
    @Kaamz1926 4 года назад +14

    2:28 he says we need to subtract red numbers from the green numbers and then he does exactly the opposite

    • @southernkatrina8161
      @southernkatrina8161 4 года назад +2

      Yep -5. I think he just said it wrong.

    • @cankarayel5063
      @cankarayel5063 3 года назад

      Determinant is 5. He did it true but he said that wrong.

  • @corpliner1506
    @corpliner1506 5 лет назад +2

    Thank you for a concrete method of finding inverse. I can follow the exact same process each time unlike row operations

  • @constantrage8878
    @constantrage8878 8 лет назад +1

    After the you get the final N matrix, how do you use it to solve for the values

  • @AsraHussain
    @AsraHussain 9 лет назад +3

    Thank you sooooooooooooooo much this helped me alot...god bless yuh!

  • @SkPnDoWnStAiRs
    @SkPnDoWnStAiRs 11 лет назад +3

    Thank you! Honestly, this is much easier for me than using Row operations

    • @woodchuk1
      @woodchuk1 Год назад

      You’re not alone…I HATE using row operations, because I can never get the numbers to give me the identity matrix. I get it right in one place and it screws up somewhere else. This method always works without any of that nonsense, even if it takes longer. The Gauss Jordan process is about as useful as a screen door on a submarine if you’re doing this by hand.

  • @parasharpanchal6857
    @parasharpanchal6857 8 лет назад +1

    man... the way u calculated the determinant was amazing ... thank u sooooo much .. i had already studied matrices in my previous semester but i had to recall it and i referred to ur video and it blew my mind.... keep on making such videos with ur own techniques.. thanks a lot buddy

  • @sppart1185
    @sppart1185 2 года назад

    Wow, you're video is quite the one... I can't believe I learnt over 5 periods in school to finally be able to find the inverse of a matrix (and still couldn't do it better) and you just did it in less than 15 minutes.... Thank you thousand times...

  • @jessicajolly7828
    @jessicajolly7828 8 лет назад +4

    Very helpful! I am taking an R class and they are asking me to do inverses' of matrices, and I had NO idea what it was.

  • @kree5423
    @kree5423 5 лет назад +4

    i had to try and fit this whole video into 1/6 of my reference sheet

  • @sebastianringlstetter6103
    @sebastianringlstetter6103 6 лет назад +1

    Bless your soul, you beautiful man. I have no idea how you made this up, but it works like a dream. Thank you so much

  • @brunosanchez6294
    @brunosanchez6294 5 лет назад

    Love how you go slow step by step even on the super simple multiplications 👏🏻

  • @TimerUchiha
    @TimerUchiha 10 лет назад +24

    It's so stupidly long and tedious, nonetheless great vid

  • @orealz
    @orealz 6 лет назад +6

    What if the determinant turns out to be a negative number?

    • @berkburakbiyikli5655
      @berkburakbiyikli5655 5 лет назад

      It will still be alright. The negativity could only show the direction so as along as the determinant is not zero, it will have an inverse.

  • @Deogharzone
    @Deogharzone 5 лет назад

    Thanks sir !
    This video is very helpful and easy to solve this problem .
    So thank you sir 🙏😊

  • @Sophiex7
    @Sophiex7 5 лет назад

    Hi, thanks for your video.
    Can you use this method to find the determinant of 4x4, 5x5 etc instead of going through all the row operations?

  • @konnichiwa8762
    @konnichiwa8762 10 лет назад +5

    Thank you very much!! This video already help me a lot with my assignment ^_^

  • @Shane8
    @Shane8 10 лет назад +6

    That is the arbitrary process I have ever seen.

  • @rishabhc32
    @rishabhc32 9 лет назад +4

    What is the name of the method you used to find inverse of the matrix?????

  • @littleguydb2
    @littleguydb2 5 лет назад

    You are a lifesaver. Its almost as if college finite teachers teach this in the most complicated way possible for no reason. always trying to make us use the damn identity matrix like nah fam yall tweakin. Thank You again

  • @cjracer1000
    @cjracer1000 8 лет назад +11

    This method is completely unintuitive, but it works and that's all that matters. You explain every step. There are no arithmetic or logical leaps made and therefore there's no ambiguity or need for question. Your videos are literally perfect. You make patrickJMT look like garbage as he skips steps and I'm forced to do arithmetic on my own as if it's trivial. Well if it's so trivial then just do it. You do it! Thank you very much for being thorough and exhaustive. As far as I'm concerned, this is the only video on this topic.

  • @rasberry272
    @rasberry272 7 лет назад +51

    This method seems super impractical

  • @hichamouazi
    @hichamouazi 9 лет назад +3


    Thank you, this is the easiest way to inverse a matrice so far I guess

  • @guilhermealves9795
    @guilhermealves9795 4 года назад

    Congratulations, you have a lot of patient and are an excellent teacher!

  • @rimberse3405
    @rimberse3405 4 года назад +10

    14:38 ahah, nice ending lmao
    Nearly destroyed my self-esteem...
    I would probably spend over 40min...

  • @Xmasparol
    @Xmasparol 10 лет назад +61

    who invented this?

    • @retroSTAAR
      @retroSTAAR 10 лет назад +23

      looool im thinking the same thing i mean what is the point

    • @RenzoIsHereYT
      @RenzoIsHereYT 6 лет назад +2

      James Bondoc in CP matices work so different

    • @jonbowman7686
      @jonbowman7686 5 лет назад +26

      In engineering, we actually use this to solve multiple dynamics equations simultaneously, so we can determine how all the different parts of a mechanism will move- like all the positions and max/min angles of all the parts in an engine for example. I had to refresh my memory of how to do this lol, that's what brought me here.

    • @glittergirl19022
      @glittergirl19022 5 лет назад

      exactly. i mean why

    • @thewarlord6529
      @thewarlord6529 5 лет назад +7

      A random genius 200 years ago, that never even thought about how much misery he was inadverdently causing among future high school students.

  • @spinmaster3983
    @spinmaster3983 5 лет назад

    In step 4 when you have to alternate the signs do you always go + - + and so on or do you use the opposite sign of the number in the first row first column? For example if the number in the first row first column was positive instead of negative would you go - + - ?

  • @udarajayawardena2018
    @udarajayawardena2018 4 года назад +1

    Clean & Clear! Thank you Math Meeting

  • @user-tx2tl4og8u
    @user-tx2tl4og8u 8 лет назад +5

    شكرا جدا انا مكنتش فاهم اجيب مرافقات المصفوفه ازاى دلوقتى بقيت اعرف و لو جت فى الامتحان مش حامل همها

  • @GreaterDeity
    @GreaterDeity 9 лет назад +51

    This method made absolutely no sense to me; seems very arbitrary, but it's right.

    • @GreaterDeity
      @GreaterDeity 9 лет назад

      ***** Ill just admit I am too stupid for this one. I tried going over it for a good 30 minutes and couldnt find the logic of the pattern.

    • @GreaterDeity
      @GreaterDeity 9 лет назад

      I do the same to find the inverse, but it gets really confusing if I use the determinant. Row ops and Gaussian Elimin. aren't that bad, as long as I don't make arithmetic errors.

  • @mikesmate6672
    @mikesmate6672 6 лет назад +1

    Thanks bro I love these steps they are fast and straight to the point, sure the computer gives the answer instantly because all these steps were incorporated into it by someone who knew them.

  • @ripfumelohlungwani2812
    @ripfumelohlungwani2812 6 лет назад +1

    thank you bro, you've just simplified this nightmare to me. it was giving me a tough time until i come across this video.

  • @AK.
    @AK. 10 лет назад +3

    what if this was a 4x4 matrix? Would I have to draw 4 diagonal lines?

    • @iohanb211
      @iohanb211 10 лет назад

      no,it does not work this way,it is a special way on 4x4 but i cant remebrer:(

    • @1ThomasTom
      @1ThomasTom 10 лет назад +3

      Iohan Kellerman You'll have to write a 4x4 matrix of which all diagonal entries from left above to right under are 1 and the non-diagonal 0. Place this matrix next to the 4x4 you want to know the inverse of. Then row reduce this new 4x8 matrix so that the left 4 columns look like the matrix you wrote next to your original matrix (with the 1's and 0's). If you've done this correctly, you'll see the inverse of your matrix in the right 4 columns of the 4x8 matrix. I think this works for every matrix that is invertible. You can try it out on the 3x3 in this clip to see if you understand.

    • @iohanb211
      @iohanb211 10 лет назад

      I see,but i don't need 4X4 matrix,my exams are how can i say a medium difficulty,i just needed to remember how it works on a 3x3!Thank you!

  • @XrGrimreap3rX
    @XrGrimreap3rX 9 лет назад +8

    What if the determinant is 0?

    • @redadrissi4460
      @redadrissi4460 6 лет назад +1

      you will have to pseudo inverse it

    • @ttvurmom6185
      @ttvurmom6185 5 лет назад

      Then you quit math class and start gambling.

  • @angelbunny225
    @angelbunny225 9 лет назад +4

    Wow thank you so much! You saved my life in this crappy maths class.

  • @SundaresanMuthuswamy
    @SundaresanMuthuswamy 2 года назад

    VERY NICELY DONE. Thank you for the clarity and approach.

  • @billtruttschel
    @billtruttschel 9 лет назад +11

    I'm really annoyed by a few things in this video. First, at 2:25, you said you were going to subtract the red numbers from the green numbers, when you clearly did it the other way around. Second, at roughly 12:00, you have two different matrices that you're calling N. Third, N itself is a matrix, so when you write [N] you are saying the matrix consisting of the matrix N, which is also wrong.

    • @ttvurmom6185
      @ttvurmom6185 5 лет назад

      Bill T I caught that too, bro.

  • @GermanSnipe14
    @GermanSnipe14 10 лет назад +39

    Lol if people are learning matrices, I'm pretty sure they know arithmetic, you don't need to waste time explaining how to subtract numbers

    • @weeeho8005
      @weeeho8005 6 лет назад +14

      Lol if people are learning matrices on youtube, I'm pretty sure they know they can skip parts of the video, you don't need to waste time explaining how to explain better

    • @myhops
      @myhops 6 лет назад +6

      I think it is necessary, because with your mindset, it's easy to assume too much and leave out parts that can be easily forgotten or overlooked.

    • @crwelch12
      @crwelch12 6 лет назад +6

      you clearly underestimate the human race's ability to fuck up

  • @soheylmoheb7273
    @soheylmoheb7273 8 лет назад

    you don't know to what extent you've helped me with your video ;-))) that was so awesome. what I couldn't keep in my mind for more than 3 years, easily stuck in my mind .
    thanks so much for your great video

  • @spmcdade901
    @spmcdade901 6 лет назад

    Thanks, that was very easy to follow. Much better than drawing loads of random triangles

  • @bh4t128
    @bh4t128 9 лет назад +3

    hard way to solve inverse of 3x3 matrix

  • @Boristien405
    @Boristien405 7 лет назад +1

    Helped me learn how to find the determinate lol. Thank you!

  • @trishamaeyurieg.camontoy852
    @trishamaeyurieg.camontoy852 2 года назад +1

    This is so much easier than doing row operations... BIG THANKS!!! I managed to do my assignment thanks to this :D

    • @theboldguy.
      @theboldguy. Год назад

      On row operations you may stuck like crazy and always go from 0's on the diagonial to positive numbers back and forth never getting a solution .....

  • @theguyaneseumbrella2762
    @theguyaneseumbrella2762 4 года назад +1

    Its 2019 and this still helped me ... thank you

  • @MadFoxRemixed
    @MadFoxRemixed 6 лет назад

    still helpful 6 years later, you rock dude!

  • @miriamburton8405
    @miriamburton8405 7 лет назад

    This is soo clear and simple, great video thanks a lot!

  • @emmanuelhoustonkwashie9222
    @emmanuelhoustonkwashie9222 5 лет назад

    This guy deserves my Professor's pay check every month for 60 years. This is the best way to teach students but some Professors take delight in failing students so they don't even care teaching it wrong. I love this video.

  • @user-rj8tr6sp7e
    @user-rj8tr6sp7e Год назад

    this was so easy to understand thank you you're a life saver !!

  • @KunwarDigvijayPratapSingh
    @KunwarDigvijayPratapSingh 8 лет назад

    It couldn't have been more easily explained. Thanks! (:

  • @sinepsinep123
    @sinepsinep123 5 лет назад

    Many thanks from Germany! Searched for a proper video which explains the whole process and in every video I watched anything missed, so I always had sign mistakes or wrong numbers.

  • @MrAZCJ
    @MrAZCJ 5 лет назад +1

    That ending tho🤣🤣🤣🤣🤣
    Sums up our education spending so much time on a problem that you can solve in a sec on a computer

  • @Surenghatang
    @Surenghatang 11 лет назад

    would this way of finding matrices would work for 2*2 matrices or 4*4 ?

  • @shantanu925
    @shantanu925 5 лет назад +1

    World needs more guys like you!

  • @brianomarion
    @brianomarion 6 лет назад

    I know its years down the line since this video, but i swear you just saved my dumb brain all the stress of not getting what our eacher was doing,

  • @XtraL0UD
    @XtraL0UD 8 лет назад

    Very comprehensive and easygoing guide, you are a god!

  • @jasmeenrahman3298
    @jasmeenrahman3298 11 лет назад

    thank you! you have been of great help for my last minute assignment submission! :)

  • @carmenwong9711
    @carmenwong9711 8 лет назад

    you taught really well and i love your last sentence

  • @oderajohana
    @oderajohana 10 лет назад

    This the best explanation of all the videos for finding inverse. thanks very much

  • @hienle1153
    @hienle1153 6 лет назад

    Very clear explanation. Wish my teacher have made it clear like you. Thank you so much.

  • @wardasiddiqui
    @wardasiddiqui 8 лет назад

    Thank you so much. This is straightforward and fun to do.

  • @terencewright2223
    @terencewright2223 3 года назад

    Great stuff, very thorough. Thank you

  • @CaityRaindrop
    @CaityRaindrop 10 лет назад

    I found this very helpful, trying to self teach myself this material for a transfer exam to change courses and the book just jumped straight into asking questions without even giving a formula.

  • @rajaninair4377
    @rajaninair4377 8 лет назад

    thank u so much.....i was very confused to do the inverse of matrix before watching this video. you explained it in an efficient way any body can understand easily. Excellent !!!!!

  • @sulangnelson7269
    @sulangnelson7269 5 лет назад

    finally..took a month for me searching for this type of video..very helpful...perfect

  • @fousin3391
    @fousin3391 2 года назад

    Only good, consistent explanation i could find

  • @ReihaAli
    @ReihaAli 6 лет назад

    Thank you soooo much! I was really struggling and you've just helped me!

  • @theodoresweger4948
    @theodoresweger4948 4 года назад

    Well done and appreciate the work put into making a good explanation

  • @JoshHarrisPOV
    @JoshHarrisPOV 6 лет назад

    I fucking love you, 6 weeks of my lecturer trying to explain it and you managed it in a 15 minute video, thank you sir.

  • @omermirza3210
    @omermirza3210 5 лет назад

    One of the most well explained tutorial over youtube

  • @topsixgaming
    @topsixgaming 2 года назад

    Thanks man. You’re genius! 🙌🏽 this is so comprehensive! 😭😭🙌🏽🙌🏽

  • @aygunvarol
    @aygunvarol 6 лет назад

    Thank you, for stating clearly this complex subject of linear algebra.

  • @andrejuntarciego7506
    @andrejuntarciego7506 6 лет назад +1

    You should've multiplied 1/determinant and the transposed matrix right?

  • @richiepanda5975
    @richiepanda5975 4 года назад

    thank you very easy to understand. Glad I came across this video. Just subbed

  • @teargun1
    @teargun1 12 лет назад

    You are awesome man.. Just made the matrix look easy... Thanks....keep uploading...

  • @khalidabuhijleh3475
    @khalidabuhijleh3475 6 лет назад

    can we use this way to find the inverse of any system n*n or just 3*3 ??

  • @bryce7355
    @bryce7355 7 лет назад +1

    Perfect. I needed a brief review. Engineering courses use linear algebra and all the good stuff with variables, so it usually is a little more pencil being pushed down. And tears, don't forget the tears...

  • @Duckington10
    @Duckington10 8 лет назад

    The part about reflecting the elements in the diagonal line is the transpose of the matrix which is where rows become columns and columns become rows.

  • @khaddyceesaydaffeh6188
    @khaddyceesaydaffeh6188 3 года назад

    This video is really a life saving one....thanks so much Sir. God bless you

  • @kayouel4551
    @kayouel4551 8 лет назад

    i have an exam tomorrow and u just helped me thank u so much, from morocco

  • @HDitzzDH
    @HDitzzDH 5 лет назад

    10:48 I think it's easier to just say that the rows becomes the columns (first row becomes the first column, second row becomes the second column etc), perhaps it makes it easier to grasp what we're doing (at least it was for me).