welcome to another video let's find the igen values and igen vectors of this 3x3 Matrix now I will not be explaining the meaning of Igan value or igen vectors I just want to do calculation it's just computation today now if you need to know what they mean I will leave the link in the description to the other videos where I broke down the meanings of those terms okay let's get into the video [Music] so our first move is to find the igen values and we just need to know that a minus Lambda I the determinant of that will be equal to zero and that's so easy we know that Aus Lambda I the determinant of this is equal to zero and what does that mean well I'm going to come back and write the lambdas so we're going to have Lambda 1 equals we're going to have Lambda 2 and we're going to have let's do it here and then we're going to have Lambda 3 okay so we're going to have all our three lambdas here but let's do the calculation here let's take this determinant the determinant of this remember we're subtracting Lambda so it's going to be 2us Lambda where we have 0 1 and then we have -1 here we have 4 - Lambda 4 - Lambda so the lambdas are subtracted along the main diagonal and then we have minus1 and this is going to be -1 2 and this is 0 - one 0 - Lambda rather so I'm just going to write minus Lambda so here we need to take this determinant and we know this has to be equal to zero okay so we're going to take this and say that two 2 - Lambda multiplied by the determinant of this 4 - Lambda - 1 2us Lambda that's the first step okay minus well zero if it use this one you don't need to bother yourself because whatever you multiply by zero is going to end up as zero that's why I chose this um row so let's pick this one it's going to be plus minus plus so it's just a plus and then I have one let's just write it times the determinant of um this is going to be Min -1 you have 4 - Lambda you have Min -1 you have two so you're watching this because you already know how to take determinants of a 3X3 Matrix if you don't well I'll leave the link in the description also and this is what we have and this is equal to zero so let's multiply this out you have 2us Lambda multiplied by the determinant of this is going to be Lambda * um 4 - Lambda - 2 * -1 is- so that's going to be + 2 we close this plus we go in here it's going to be um what would this be this is going to be -2 minus which is going to be plus because this times this would be this and you have 4 minus Lambda okay equals z let's simplify a bunch of algebra remember it's this multiply this minus this product so if I try to simplify each of the parentheses I'm going to get 2 - Lambda I have here I have I'm going to end up with Lambda 2 - 4 Lambda + 2 and here I have plus what would this be I'm going to have um plus 2 oh it's 2 - Lambda oh looks like there's a 2us Lambda here too so I can easily Factor these so I have 2us Lambda and then on the inside I have there's going to be a one left here add that one to this two when you remove the parentheses it's going to become Lambda 2 - 4 Lambda + 3 equal 0 nice now what is my here I know that this is now a cubic equation but it's it's in factored form so if I write it out it's going to be 2 minus Lambda if I factor this this is going to be Lambda minus 1 oh and Lambda minus 3 that's going to be zero so that tells me that my Lambda 1 is two whatever makes this zero so I have two I got one I got three oh nice I can write this as one two and three so that's the process remember taking the determinant is the hardest part once you take the determinant you can solve the equation you get at the end and you get your igon values so we have obtained the IG values for this and all we did was subtract Lambda from each of these main diagonal entries and take the determinant which we did here and this is where we are so now let's go find the igen vectors because I'm going to give you some exercises at the end of this video I am just going to do the calculation for the igen vectors when Lambda is equal to 1 now for Lambda equals 2 and Lambda equals 3 I'll just tell you what the answers are at the end of the video but I'll need you to do that yourself or maybe I shouldn't tell you the answer well I'll decide that at the end okay so how do you get the igen vectors igen Vector corresponding to Lambda equals Lambda equal 1 well we're going to say when Lambda is equal to 1 do this look we go back to the Matrix we're going to be subtracting one from each of these okay so we know that a minus Lambda I that thing we used here instead of finding the determinant we just want the matrix it's going to be 2 - 1 is going to let's write it 2 - 1 0 1 we're going to have -1 4 -1 -1 and then we're going to have -1 2 0 -1 so this is what we have and we know that whenever you multiply this by The Matrix I think I have to rewrite this so if we multiply this by the vectors X1 X2 X3 by the vector this Vector you're going to always get zero and that is from the equation I need to erase this and that is from the basic definition that a minus Lambda I multiplied by the igen vector of Lambda is equal to zero so what we have here is this Matrix this is the V we're looking for and you will always get the zero Vector now we're looking for the V corresponding to this Lambda so if I call this Lambda 1 this is V1 so you're going to get three of them since we have three different lambdas okay so now let's do this calculation let's resolve this so what we have so say Lambda equal 1 so we have um this is going to be one so the Matrix we have actually looks like this we have 1 zero and one and then we have -1 3 and -1 -1 3 -1 and then we have -1 2 -1 - 1 2 - 1 by X1 X2 X3 is equal to 0 0 0 my recommendation is you solve a system of equations but this is very easy to solve because what you're saying is that you see when you multiply this way this is 1 * X1 0 * X2 1 * X3 so the X2 is zero so it will not show up what you know is that 1 * X1 X1 plus the last one which is X3 is equal to Zer and what does this mean it simply means that X1 is X3 or you can say X3 let's put it this way X3 is equal tox1 let's write everything in terms of X1 okay it's just better so you start with a one you don't start with a zero it doesn't matter what you choose okay just keep going now if I go to the second one I know that X1 + 3 X2 - X3 is equal to Z but I already said that X3 is X1 so I'm going to go here and replace X3 with X1 you see that but if I put X1 is going to become positive X1 but positive X1 - X1 tells me that 3x2 equal 0 which implies X2 is equal to Z so now I have found X2 at this point this is where many students get confused do not try to solve to know what X1 is because nobody knows don't try to find X2 but unfortunately we were just trying to find x2 in terms of X1 but it turns out that X2 will always be zero are we done yes because now we know that whenever we know we just choose something we can choose for X2 X2 will always be zero but we can choose for X1 and we know what X3 is going to be we can choose for X3 we know what X2 is going to be so all you're saying is I Vector correct corresponding to Lambda 1 = 1 is you want to know what that is I'm going to write it here it is this Vector X1 X2 X3 we know X2 is always zero no matter how you try what will X3 be we don't know what is X1 going to be we don't know but typically you just choose one okay I'm going to say whenever X1 is 1 X3 is going to be minus one so let X1 be 1 let X1 be minus one now in this case don't choose zero because if you choose zero here it means that everything is going to be zero and an IG Vector can never be the zero Vector so you can't do that now you can choose two you can choose seven it doesn't matter but it just makes sense to choose one because one is easy to manipulate so that's why we fix this to be one pick this to be minus one now you'll ask me what if I say X1 is minus one that's fine nobody's nobody's troubling you just make sure you keep your order you're free to choose this cuz again if I multiply this by minus one it doesn't change the igon vector that's your answer so remember the mission is not to solve for X1 X2 X3 is just to be able to write x2 in terms of X1 X3 in terms ter of X1 okay and some people do it the other way but just write all of them in terms of one of the um components and then use it to form your igen vector there's nothing else to do do the same thing when Lambda equals 2 and when Lambda equals 3 and I'll see you in the next video okay let me tell you what the answer to the other ones are in case you want to check I'll leave it in the description never stop learning those who stop learning stop living bye-bye