in this video we're going to talk about how to determine the inverse of a three by three Matrix so let's say if we have Matrix a and it contains the elements 1 2 negative one negative two zero one negative 1 and 0. so go ahead and determine the inverse of Matrix a so first we need to rewrite this in the form of an augmented Matrix with Matrix a and the multiplicative identity of a three by three Matrix so here's Matrix a it's one two negative 1 and so forth and the multiplicative identity Matrix is going to be one zero zero zero one zero zero zero one so now here's what we need to do we need to perform Elementary row operations and basically take the left side and make it look like the right side now the operations that we apply to the left side we have to apply to the right side so the right side is going to change as a result and whatever the result is on the right side that is going to be the inverse of Matrix a so let's begin by turning this number into a zero to do that let's subtract Row 1 by Row three and let's apply the result to Row three now Row 1 is not going to change so let's rewrite that Row 2 is also not going to change as well now for Row three it's going to be 1 minus 1 so that's zero and then it's 2 minus negative 1 or 2 plus 1 which is three and then we're going to have negative 1 minus zero so that's negative one and then 1 minus zero is one zero minus zero is zero and then zero minus one is negative one so now what do you think we need to do next how can we turn this negative 2 into a zero what row operations do we need to apply to row two to do so so we're going to say 2 times R1 plus R2 let's do that so we'll change in row two Row one is going to stay the same Row three will remain the same as well so two times one plus negative two that's going to be zero two times two plus zero that's going to be four two times negative 1 which is negative two plus one negative one two times one plus zero is going to be 2 2 times 0 plus 1 that's 1 2 times 0 plus 0 is 0. now let's turn that number into a zero so we need to apply the row operation to Row three I'm going to multiply R2 by 3 because 3 times 4 is 12. and then I'm going to subtract it by 4 times R3 because negative 4 times 3 is negative 12. and they'll cancel to zero so Row one is going to stay the same and let's rewrite row 2. so we're going to have 3 times 0 minus 4 times 0. which is zero and then 3 times 4 minus four times three that's going to be 0 as well and then 3 times negative 1 which is negative three minus 4 times negative one so we have negative 3 plus 4. that's going to equal to 1. and then 3 times 2 is 6 minus 4 times 1 so 6 minus four is two three times one minus four times zero that's three and three times zero minus four times negative one is four now what we need to do is turn this number into a zero so that's going to be straightforward we just got to add Row one and Row three to accomplish that so let's rewrite rows two and three so it's going to be one plus zero which is one and then two plus zero that's two negative one plus one is zero and then one plus two is three and then zero plus three is three and zero plus four is four now let's focus on that number what row operations do we need to apply to row two in order to make it zero all we need to do here now is add row two and Row three so notice what's going to happen so let's rewrite Row one now let's rewrite Row three so row two plus Row three zero plus zero is zero and then four plus zero is four negative one plus one that's going to be zero and then we have two plus two which is four and one plus three that's four and then zero plus four is also four now what we need to do at this point wait hold on I didn't copy this correctly this is supposed to be one 2 0 and 3 3 4. so everything else should be fine at this point this is what I should have right now so at this point we need to turn this number into a zero so we need to apply the operation to row one so let's take two times R1 and subtract it by R2 so we're going to have 0 0 1 two three four and then 0 4 0 4 4. now we're going to take two times one minus zero so that's going to stay 1. and then 2 times 2 minus four that is now zero two times zero minus zero is zero and then two times three which is six minus four that's two the next one is going to be the same 2 times 4 is 8 minus 4 is 4. and one thing I do need to fix two times one minus zero that's supposed to be two and uh have a one here so let's fix that now we need to do is multiply the first row by one half and the second row by one over four and that should give us what we need so half of two is one and half of 4 is 2. one fourth of four is one so all of these will be one and then the last one is not going to change so notice that here we have the multiplicative identity Matrix i3 and this side represents the inverse of Matrix a one one two one one one two three four now to confirm that it's indeed the inverse what we need to do is multiply Matrix a by the inverse of a and show that it's equal to i3 so in other words we need to take Matrix a which was 1 2 negative one negative 2 0 1 one negative 1 0 and multiply it by the inverse of that Matrix which is one one two and one one one two three four and if this is the Matrix of a if we did it correctly we should get this answer one zero zero zero one zero zero zero one so let's find out if we did it correctly so I'm going to put my answers in here first we need to take the first row and multiply by the first column and add the products so it's going to be 1 times 1 so we're multiplying these two first and then it's going to be plus two times one that's these two and then plus negative 1 times 2. so 1 times 1 is 1 2 times 1 is 2 negative one times two is negative two these two cancel and so we get one so we have the first entry now let's multiply Row 1 by column two so it's going to be 1 times 1 plus 2 times 1 plus negative one times three so this is one plus two minus three one plus two is three three minus three is zero now let's try Row one by column three so that's one times two plus two times one plus negative one times four so this is two two minus four two plus two is four four minus four is zero now let's move on to row two and let's multiply it by column one so it's going to be negative 2 times 1 plus 0 times 1 plus 1 times 2. so this is negative two plus zero plus two negative two plus two is zero now let's take row two and multiply it by column two so it's going to be negative two times one plus zero times one plus one times three so that's negative two plus zero plus three negative two plus three is one now let's multiply Row Two by column three so negative two times two plus zero times one plus one times four so this is going to be negative four plus zero plus four which adds up to zero now Row three times column one let's try that so it's going to be one times one plus negative one times one plus zero times two so it's 1 minus one plus zero which is going to be zero next is going to be Row three and then column two so this takes some time I mean it's pretty laborious but that's how you can confirm it so it's going to be one times one plus negative 1 times 1 and then 0 times 3. so this is one minus one plus zero which is zero and finally the last one it's going to be Row 3 times column three so that's going to be 1 times 2 plus negative 1 times 1 plus 0 times 4. so it's 2 minus one plus zero two minus 1 is 1. and so we do get the identity Matrix so that tells us that this is indeed the inverse of a so now you know how to find the inverse of a three by three Matrix thanks for watching