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let's explore what happens to determinants when you multiply them by a scalar so let's say we wanted to find the determinant of this matrix of a b c d by definition the determinant here is going to be equal to a times D minus B times C or C times B either way ad minus BC that's the determinant right there now what if we were to multiply one of these rows by a scaler let's say we multiply it by K so we have the situation a B let's multiply the second row times K so we have K C and K D now what it's what's the determinant going to look like we're going to have a times K D or just write that as K ad minus K C times B or we could write that as K BC if we factor out the K we get that equals K times ad minus BC now you immediately see that this thing is the same thing as this thing so this is equal to K times the determinant of a b c d so when you just multiply a row by a scaler just one row not the entire matrix when you just multiply a row by tum by some scalar the resulting determinant will be the original determinant times that scalar now you might say well what happens if I multiply if I multiply the whole matrix times that scalar well that's equivalent to multiplying by a scalar twice right if I say that let's say I have the matrix a and the matrix a is equal to a b c d if i were to think about the matrix K a now I'm not just multiplying one row I'm multiplying the whole matrix by a scalar this is going to be equal to K lowercase a KB K C and K D and when you figure out its determinant the determinant the determinant of K times a is going to be equal to the determinant of K a KB KC kay C and KD and here you're immediately going to see her to end up with K squared terms you're gonna have K squared times ad it's going to be equal to K squared ad minus K squared B C or K squared times ad minus BC or K squared times the determinant of just of just a so you have to be very careful and this is only for a two by two case you'll find out if this was an N by n matrix that this would have been K to the N so the takeaway is the only way you can say that it's going to be a some scalar multiple times your original determinant is only if you multiply one row one row times that scalar multiple not the whole matrix let's see how this extends to let's extend it to maybe a three by three case let's take it do it and you know you might say hey Sal you just pick the second row does it work with the first row I'll leave I'll leave that for you to determine but it does it does work it doesn't matter which row I multiplied it by let's take the three by three case let's say we have some matrix let's call this a again I'm redefining a going to be a b c d e f g h i and then if you take its determinant let's just take its determinant the determinant of a is going to be equal to well we could do it a couple of different ways but i'll just pick some arbitrary row because that's the row that we're going to multiply by some scalar so let's just take that role right there so determinant of a is going to be equal to remember the plus/minus pattern right remember plus minus plus minus plus minus plus minus plus that little checkerboard pattern so d is a minus right there so it's going to be equal to minus D times the determinant of its sub-matrix so you cross out that column and that roads B CH i b c h i and it's going to be plus e times its sub-matrix a/c g i a/c g i and it's going to be minus F times you get rid of that row that de gh the determinant of de gh that's the determinant of this matrix a now what if we define some new matrix here let's call it a prime let me may scroll down a little bit let me define a prime right here a prime I'm just going to multiply this row by a scalar so it's going to be equal to a b c KD ke + KF I'm not multiplying the whole matrix times the scalar I can't say this is ka I'm just multiplying one of its rows and then I have G H and I so what's the determinant of a prime going to be the determinant of a prime and I put that Prime data so it's different than a or maybe it's you know but it's derived from a I just multiplied one row of a times a scalar well I can go along that same row that I did up here I go along that same row and the only difference is is that instead of having a d I now have a KD instead of an e I now have a ke so instead of a d I'm gonna have a KD there instead of an e I'm gonna have a ke there so I can it's going to be this exact same thing but I can replace this guy this guy in this guy with them multiplied by K so it's going to be equal to minus KD times the determinant of the sub-matrix BCH I I'm not even going to look over here it's going to be the same thing as that one up there plus ke times the determinant of AC GI plus minus minus KF times the determinant of de gh and what is this equal to this is equal to if you just factor out the K it's equal to K times this so it's equal to K times the determinant of a so our result also worked for three by three case I just happen to pick the middle row but I encourage you to pick other rows and to see what happens and so let's actually do it for the general case because I've just been giving you particular examples and I like to show you the general proof when the general proof isn't too hairy so let's say I have an N by n matrix so let's say that I have a matrix a let's say that a is n by n so it equals you can write it like this this is the first row first column a 1 1 a 1 2 all the way to a 1n I'm going to pick some arbitrary row here that I'm going to end up multiplying by a scalar so you know we could go down here let's say row a I this is a I won a I 2 all the way to a I n this is some row that I'm going to use to determine the determinant remember we can go down any row to get the determinant then finally you keep going you get a an1 an2 all the way to a NN this says this is as general as you can get for an N by n matrix now let's figure out its determinant so the determinant of a now I'm just going to go down this row right there that row right there so the determinant of a the determinant of a is equal to what it's equal to well we have to remember the checkerboard pattern and we don't know where we are in the checkerboard pattern because I just picked an arbitrary general row here but we can use the general formula that the sign is going to be determined by negative 1 to the I I don't know if I is even or odd so it's going to be I plus 4 this term 1 power that's its sign this is what gives us the checkerboard pattern let me make that clear it looks complicated but this is just a checkerboard checkerboard checkerboard pattern that's that's just that right there times this term right there so times a I so the coefficient a a I 1 and then times this guy's sub matrix and you remember the sub matrix you get rid of this row and this column is going to be everything that's left over so times that sub matrix is of AI 1 AI 1 and then you're going to keep and there's going to be plus let me just keep doing it plus negative 1 to the I plus 2 times a I to times its sub-matrix I 2 all the way you just keep going plus minus one to the I plus n times AI you're in the nth column and then its sub-matrix this is going to be an N minus 1 by n minus 1 matrix all of these are going to be I n just like that that's the determinant of a and we can actually rewrite it in Sigma notation that'll simplify things a little bit so the determinant of a we can rewrite it as the sum the sum from J is equal to 1 to J I'll write it explicitly here J is equal to n of of negative 1 to the I plus J times a I J and then each of the sub matrices a I J this thing right here this thing right here is just another way it's just another way of writing this thing I wrote up there where you know I'm just saying the sum you just take J equal 1 put them in there you get this term right there you take J equal 2 you add it you get this term right there you keep doing it you get J equal and you get that term right there so these are these two things are equivalent so what happens what happens if I have some new matrix let's say let me copy and paste my current matrix so let me copy and paste it actually let me copy and paste everything let me copy and paste everything that'll make things move quickly I copied it and now let me paste it just like that let me define a new matrix let me define my new matrix a prime it's still an N by n matrix but that row that I just happen to use to determine my determinant I'm going to multiply it by a scalar K so it's K AI k k a 1 k AI 2 k AI n just like that so what's the determinant of a prime well we're just going to go down this row again but now instead of just an AI 1 we have a kai1 instead of an AI 2 we have a kai2 instead of a K AI and we have a K AI n so it's determinant is just going to be this same thing but we're gonna have instead of an AI J everywhere we're gonna have a que AI J so this is going to be since the determinant of a prime and so this is equal to this is equal to we could just take out this this constant right here it has no eyes or J's in it so we can just has no J's in it in particular so we can just take it out so it's equal to K times the sum from J is equal to 1 to J is equal to n of minus 1 to the I plus J times a IJ a and this is the coefficient and then this is the sub matrix for each of those coefficients a IJ that's a matrix right there an n minus 1 by n minus 1 matrix and then you know you immediately recognize I think you saw where this was going this right here is just the determinant of a so the determinant so we get the result that the determinant of a prime is equal to K times the determinant of a times the determinant of a so we've just shown you in general you have any N by n matrix if you multiply only one row not the whole matrix only one row by some scalar multiple K it's the resulting determinant will be your original determinant times K now as I touched on this in the original video what happens if you keep what if you multiply what is the determinant what is the determinant of K times a so now we're multiplying every row times K or another way to think about is you're multiplying now I guess you're multiplying n rows times K right so you're doing this n times so if you multiply K times itself n times will you get you get K to the N so this is going to be equal to K to the N times the determinant of a right if you just do it once you get K times the determinant of a now if you do a 2nd row you're going to get K times K times the determinant of a you do a third row you can get K to the third time's the determinant of a the fourth row K to the four times the determinant of a if you do them all all n rows you're gonna have K to the N times the determinant of a anyway hopefully you found that interesting and I encourage you to experiment with this other try-try going down a column and seeing what happens