Simpler 4x4 determinant

I have this 4x4 matrix a here and let's see if we can figure out its determinant the determinant of a and before just doing it the way we've done it in the past where you go down one of the rows or one of the columns and you notice there's no zeros here so there's no easy row or easy column to take the determinant to buy you know we could have gone down this row and do all the sub matrices but this becomes pretty involved you end up doing 4 3 by 3 determinants and then that's complete each of those are composed of 3 2 by 2 determinants becomes a pretty hairy process let's see if we can use some of the realizations we've discovered in the last few videos to simplify this process well one of the realizations is that row operations or if you subtract let me write it this way if you replace if you replace Row row J with let's say row J minus some scalar multiple times row row I it does not change the determinant does not change the determinant of a we saw that I think it was two videos ago so this was a pretty big realization we can do these type of row operations that it won't change the determinant the other realization we had was that these upper triangular matrices upper triangular matrices you can figure out their determinants so what is upper triangular look like when we just review it the upper triangular everything below the diagonal so let's say the diagonal has let me just draw it terms like that these are some nonzero terms so they don't have to be then upper triangle upper triangular everything below the diagonal is a 0 everything below the diagonal is a 0 and everything above the diagonal mate probably isn't a 0 but you never know but they're nonzero terms so all the red stuff here is nonzero all this stuff in green is 0 I didn't touch out in that video but there is also such a thing as a lower triangular lower triangular that you might have guessed how it looks everything above the main diagonal is 0 so this is the main diagonal right here all the way down like that although these guys are going to be non-zero although that's going to be nonzero and then the zeros are going to be above the try above the diagonal like that we saw in the last video the determinant of this guy is just equal to the product of the diagonal entries which is a very good or it's a very simple way of finding a determinant and you could use the same argument we made in the last video to say that the same is true of the lower triangular matrix that it's determinant is also just the product of those entries I won't prove it here but you can use the exact same argument you used in the video that I just did on the upper triangular so given this that the determinant of this is just the product of those guys and then I can perform row operations on this guy and not change the determinant maybe a simpler way to calculate this determinant is to get this guy into an upper triangular form and then just multiply the entries down the diagonal so let's do that so we want to find the determinant of a so the determinant of a we rewrite a right here it's 1 2 2 1 1 1 2 4 2 2 7 5 2 minus 1 4 minus 6 3 now let's put this in let's try to get this into upper triangular form so if we let's replace the second row with the second so I'm just going to keep the first row the same 1 2 2 1 and let's replace the second row the second row with the second row minus the first row the second row minus the first row is going to be equal to 1 minus 1 is 0 so in this case the constant is just 1 so 1 minus 1 is 0 2 minus 2 is 0 4 minus 2 is 2 2 minus 1 is 1 now let's replace the let's replace the third row with the third row minus 2 times the second row so 2 minus 2 times 1 is 0 7 minus 2 times 2 is 3 5 minus 2 times - is 1 2 minus 2 times 1 is 0 and then let's let me get a good color here pink let's replace let's replace the last row with the last row essentially plus the first row you can say - minus 1 times the first row is the same thing as the last row plus the first row so minus 1 plus 1 is 0 4 plus 2 is 6 minus 6 plus 2 minus 6 plus 2 is minus 4 and then 3 plus 1 is 4 is 4 so there we have it like that and this guy has 2 zeros here so maybe I want to swap some rows so let me swap some rows so if we swap rows what happens so let me write I'm gonna swap the middle two rows just for fun well not just for fun I don't because I want a pivot entry right here or I shouldn't say pivot entry I want to do it in upper triangular form so I want a nonzero entry here this is a 0 so I'm going to move this guy down so I'm going to keep the top row the same 1 2 2 1 let me keep the bottom row the same 0 0 6 - 4 4 and I'm going to swap these guys right here so this is going to be 0 3 1 0 and then 0 0 2 1 now can I just swap entries like that well I can but you have to remember that when you swap entries your resulting determinant is going to be the negative of your original determine so if we swap these two guys the determinant of this is going to be the negative the negative of this determinant when you swap two rows you just flip the sign of the determinant we saw that that was one of the first videos we did on these kind of messing with the determinants now what do we want to do here to get this this guy into upper triangular form to get this an upper triangular form it would be nice to get this to be a 0 so to get that to be a 0 let me keep everything else the same so I have a 1 2 to 1 I have a 0 3 1 0 the third row is 0 zero to one and now to /ro let me replace it with the last row minus three times this row so let me write it like this well they have to carry that negative sign as well so I'm going to replace this last row with the last row minus three times or this last row minus two times the second row I want to zero it out so 0 minus 2 times 0 is 0 6 minus 2 times 3 is 0 minus 4 minus 2 times 1 is minus 6 and then 4 minus 2 times 0 is just 4 we're almost there now we want a 0 what do we want to do we want to zero this guy out so let's replace this one so I'm going to keep my top three rows the same again and let me see if I can write it a little bit neater so my first row is 1 2 2 1 my second row is 0 3 1 0 fourth row is 0 0 2 1 and I'm going to take the matrix I haven't written the fourth row yet and of course this is going the negative of this is going to be the determinant of our original matrix because we had swapped those rows but let's replace this last row let's replace the last row with the last row plus 3 times 3 times the third row so we get 0 plus 3 times 0 is 0 0 plus 3 times 0 is 0 minus 6 plus 3 times 2 is 0 4 plus 3 times 1 is 7 and just like that we have a determinant of a matrix and upper triangular form so this is going to be equal to this is going to be equal to the product of these guys so this is going to be equal to can't forget our negative sign let's throw our negative sign out there and put a parenthesis just like this is going to be the product of that diagonal entry 1 times 3 times 3 times 2 times 7 which is 6 times 7 which is 42 42 so the determinant of this matrix is minus 42 which was pretty fast this was a pretty fast shortcut it actually turns out it tends to be computationally more efficient to use these takeaways to put things into upper triangular form first and then you know if you do swaps you have to remember to make the determinant negative and then just multiply down the diagonal we did that there we got the determinant as being minus 42