Simpler 4x4 determinant Calculating a 4x4 determinant by putting in in upper triangular form first.
Simpler 4x4 determinant
- I have this 4 by 4 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 0's here, so there's no easy row
- or easy column to take the determinant by.
- We could have gone down this row and do all the
- submatrices, but this becomes pretty involved.
- You end up doing four 3 by 3 determinants, and then each of
- those are composed of three 2 by 2 determinants.
- It becomes a pretty hairy process.
- Let's see if we can use some of the realizations we've
- discovered in the last two 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 row
- j with, let's say, row j minus some scale or multiple, times
- row i, it does not change the determinant of A.
- We saw that, I think it was two videos ago.
- So this is a pretty big realization.
- We can do these type of row operations and it won't change
- the determinant.
- The other realization we had was that these upper
- triangular matrices, you can figure out their determinant.
- So what does upper triangular look like?
- Let me just review it.
- The upper triangular-- everything below the diagonal.
- So let's say the diagonal has-- let me just draw its
- terms like that.
- These are some non-zero terms. Oh, they don't have to be.
- Then upper triangular, everything below the diagonal
- is a 0, and everything above the diagonal probably isn't a
- 0, but you never know.
- But they're non-zero terms, so all the red stuff here is
- non-zero, all this stuff in green is 0.
- And I didn't touch on it in that video, but there is also
- such a thing as a 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.
- All of these guys are going to be non-zero.
- All of that's going to be non-zero, and then the 0's are
- going to be above the diagonal, like that.
- We saw in the last video that the determinant of this guy is
- just equal to the product of the diagonal entries, which is
- 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 its 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
- So given this, that the determinant of this is just
- the product of those guys, and that 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.
- Let me 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 try to get this into upper triangular form.
- So let's replace the second row with the-- so I'm just
- going to keep the first row the same.
- 1, 2, 2, 1.
- And let's replace 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 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 2 is 1.
- 2 minus 2 times 1 is 0.
- Let me get a good color here.
- I'll do pink.
- Let's replace the last row with the last row,
- essentially, plus the first row.
- You could say minus 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 is minus 4.
- And then 3 plus 1 is 4.
- So there we have it like that.
- And this guy has two 0's here, so maybe I want
- to swap some rows.
- So let me swap some rows.
- So if we swap rows, what happens?
- I'm going to swap the middle two rows just for fun.
- Well, not just for fun.
- Because I want a pivot entry right here.
- I shouldn't say a pivot entry.
- I want to do it in upper triangular form.
- So I want a non-zero 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.
- I'm going to keep the bottom row the same.
- 0, 0, 6, minus 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 determinant.
- So if we swap these two guys, the determinant of this is
- going to be the negative of this determinant.
- When you swap two rows, you just flip the sign of the
- 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 guy into upper triangular form, it'd 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, 2, 1.
- I have a 0, 3, 1, 0.
- The third row is 0, 0, 2, 1.
- And now this last row, let me replace it with the last row
- minus 3 times this row.
- So let me write it like this.
- I have to carry that negative sign as well.
- So I'm going to replace this last row with the last row
- minus 2 times the second row.
- 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 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, the negative of this is going to be
- determinant of our original matrix, because we had swapped
- those rows.
- But let's replace this last row with the last row, plus 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 in
- upper triangular form.
- So this is going to be equal to the product of these guys.
- We can't forget our negative sign.
- Let's throw our negative sign out there and put a
- parentheses just like that.
- 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.
- So the determinant of this matrix is minus 42, which was
- pretty fast. This was a pretty fast shortcut.
- And 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.
- And we did that there, and we got the determinant
- as being minus 42.
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This is great, I finally understand quadratic functions!
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