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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.

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  • male robot hal style avatar for user migalhas1998
    Does this only work for 3x3 matrices?
    (57 votes)
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  • male robot donald style avatar for user Nikhil Prabhu
    Determinants are valid only for square matrices right?
    (39 votes)
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  • purple pi purple style avatar for user Jimmie Hill
    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)
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  • spunky sam green style avatar for user John
    what is a checker board pattern? Sal mentions it at
    (14 votes)
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  • piceratops ultimate style avatar for user Saraph
    So for 4x4 matrix, checker board patern would be + - + - and for 5x5 it would be + - + - +?
    (12 votes)
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  • leafers ultimate style avatar for user solo1118
    Why does he say "AS A HINT"?
    (15 votes)
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  • blobby green style avatar for user chanwoo1143
    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)
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    • male robot hal style avatar for user ben.samuel
      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)
  • blobby green style avatar for user jamie_chu78
    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)
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    • leaf blue style avatar for user Matthew Daly
      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)
  • leafers seed style avatar for user Ahmed Touil
    In my school the sub-matrix extraction starts from rows rather than columns, are they the same operation ?
    (2 votes)
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  • blobby green style avatar for user jacob.930321
    -(-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)
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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.