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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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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.