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## Algebra (all content)

### Unit 20: Lesson 15

Determinants & inverses of large matrices- Determinant of a 3x3 matrix: standard method (1 of 2)
- Determinant of a 3x3 matrix: shortcut method (2 of 2)
- Determinant of a 3x3 matrix
- Inverting a 3x3 matrix using Gaussian elimination
- Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix
- Inverting a 3x3 matrix using determinants Part 2: Adjugate matrix
- Inverse of a 3x3 matrix

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# Determinant of a 3x3 matrix: standard method (1 of 2)

Sal shows the standard method for finding the determinant of a 3x3 matrix. Created by Sal Khan.

## Want to join the conversation?

- Does this only work for 3x3 matrices?(57 votes)
- The method described in this video works on square matrices of any size. The shortcut method from the next video only works on 3x3 matrices.(102 votes)

- Determinants are valid only for square matrices right?(39 votes)
- Yes, you can only calculate the determinant for a square matrix.(33 votes)

- when you choose the row you will use for this method, can it be any row? For example in in your example could you use -2, 0, 0.(17 votes)
- Yes you can! Actually using that row would be a handy trick to calculate the determinant a little faster.(31 votes)

- what is a checker board pattern? Sal mentions it at0:20(14 votes)
- A checkerboard pattern is when every other square is black or white both in the horizontal and vertical directions. With numbers instead of colors, it could look like this:

1 0 1

0 1 0

1 0 1(27 votes)

- So for 4x4 matrix, checker board patern would be + - + - and for 5x5 it would be + - + - +?(12 votes)
- Yeah it's (-1)^(i+j). so if I+J is even you get positive, i + j is odd you get a negative number. So 1 + 1 = 2. (-1)^2 = 1(1 vote)

- Why does he say "AS A HINT"?(15 votes)
- Yeah, I thought so too. I found a nice intro here, everyone should watch this first.

https://www.youtube.com/watch?v=0c7dt2SQfLw(5 votes)

- CAN anyone tell me how this method gives the same result as the method 1 in the previous video??

Why is Sal not explaining this??(6 votes)- That's a fair question. I can speak with any authority on why they didn't explain it, but I suspect a proper proof of this stuff is just tricky to condense to a short video, or maybe they haven't gotten to it yet.

If you want a less rigorous proof that shows that they both work, you can do it yourself.

Write out a matrix:`a b c`

d e f

g h i

The just calculate the determinant of that using both methods, and satisfy yourself that they are both the same. It'll take a few minutes, but it's a worthwhile exercise.(4 votes)

- Does anyone know if there's any videos which go behind the intuition for this? I understood how the operation were defined right up until this section, and than bam, everything falls to pieces, nothing makes sence.(3 votes)
- I'm a big fan of finding the inherent a natural beauty of mathematics, but unfortunately explaining the determinant or arbitrary square matrices is confounding. I took a graduate level abstract linear algebra course in college where the professor wrote the most mathematically dense textbook I have ever encountered, and even he said that we would have to wait for Volume 2 before the determinant would make any sort of sense to us.

The one critical thing to take away from determinants is that if the determinant of a matrix is zero, then the matrix cannot be inverted. If you dive into the linear algebra module (and you're more than able to handle it), you can see that this makes sense because a determinant of zero means that the row vectors are linearly dependent and therefore cannot span the entire space (but if you haven't gone into the linear algebra module yet, even that is gibberish). ^_^(5 votes)

- In my school the sub-matrix extraction starts from rows rather than columns, are they the same operation ?(2 votes)
- Yes, you will proceed in exactly the same way and get the same answer. Just tilt your head when you watch Sal's videos. ^_^(5 votes)

- -(-1) is +1. And If there is -4 instead of 4 in a11, then you would have had -(-4) which is 4. So why bother with the + and – sign? Sal could have said 4*det(5,3,0,0) + 1*det(4,3,-2,0) + 1*det(4,5,-2,0)(2 votes)

## Video transcript

As a hint, I will take the
determinant of another 3 by 3 matrix. But it's the exact same
process for the 3 by 3 matrix that you're trying to
find the determinant of. So here is matrix A.
Here, it's these digits. This is a 3 by 3 matrix. And now let's evaluate
its determinant. So what we have to remember
is a checkerboard pattern when we think of
3 by 3 matrices: positive, negative, positive. So first we're going to
take positive 1 times 4. So we could just
write plus 4 times 4, the determinant of 4 submatrix. And when you say,
what's the submatrix? Well, get rid of the column
for that digit, and the row, and then the submatrix
is what's left over. So we'll take the
determinant of its submatrix. So it's 5, 3, 0, 0. Then we move on to the
second item in this row, in this top row. But the checkerboard
pattern says we're going to take
the negative of it. So it's going to be
negative of negative 1-- let me do that in a slightly
different color-- of negative 1 times the determinant
of its submatrix. You get rid of this
row, and this column. You're left with 4,
3, negative 2, 0. And then finally, you
have positive again. Positive times 1. This 1 right over here. Let me put the positive
in that same blue color. So positive 1, or plus
1 or positive 1 times 1. Really the negative is where
it got a little confusing on this middle term. But positive 1 times 1 times the
determinant of its submatrix. So it's submatrix is
this right over here. You get rid of the row,
get rid of the column 4, 5, negative 2, 0. So now we just have to evaluate
these 2 by 2 determinants. So the determinant
right over here is going to be 5 times
0 minus 3 times 0. And all of that is going
to be multiplied times 4. Well this is going
to be 0 minus 0. So this is all just a 0. So 4 times 0 is just a 0. So this all simplifies to 0. Now let's do this term. We get negative negative 1. So that's positive 1. So let me just make
these positive. Positive 1, or we
could just write plus. Let me just write it here. So positive 1 times
4 times 0 is 0. So 4 times 0 minus
3 times negative 2. 3 times negative
2 is negative 6. So you have 4-- oh, sorry,
you have 0 minus negative 6, which is positive 6. Positive 6 times 1 is just 6. So you have plus 6. And then finally you have
this last determinant. You have-- so it's going
to be plus 1 times 4 times 0 minus 5 times negative 2. So this is going to be
equal to-- it's just going to be equal with--
1 times anything is just the same thing. 4 times 0 is 0. And then 5 times negative
2 is negative 10. But we're going to
subtract a negative 10. So you get positive 10. So this just simplifies
to 10, positive 10. So you're left with,
let me be clear. This is 0, all of this
simplifies to plus 6, and all of this
simplifies to plus 10. And so you are left with, if
you add these up, 6 plus 10 is equal to 16. So the trick here
is to just make sure you remember the
checkerboard pattern, and you don't mess up with all
the negative numbers and all of the multiplying.