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## Algebra (all content)

### Course: Algebra (all content)ย >ย 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: shortcut method (2 of 2)

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

## Want to join the conversation?

- What is the determinant used for exactly?(49 votes)
- It can be used to solve systems of equations with cramers rule. It can be used to represent the cross product (a type of vector multiplication). But I believe there is more to it.(47 votes)

- at1:09what happens to the 4 and the 5 that was left out from the diagonals?(18 votes)
- The reason we copy those columns is just for visual simplicity. What's really happening is that the diagonals are wrapping around, like in Pac Man.

So the 4 is actually being used by the blue diagonal starting at 1 and the orange diagonal starting at -1. Likewise, the 5 that seems to be unused is really the 5 that is right in the middle of the matrix.(47 votes)

- Is this method applicable to any other matrices such as a 4x4 matrix?(17 votes)
- No, to find the determinant of a 4x4 matrix and greater you have to use co-factor expansion.(26 votes)

- do we have an excercise practice for calculating the determinant?(7 votes)
- Someone was listening, because it's up now!(10 votes)

- I though he said you subtracted the result of the first set of diagonals from the second? So shouldn't the answer be -4, as 6-10=-4?(7 votes)
- the actual value from the second set of diagonals is -10. So you are subtracting a negative which is the same as adding the positive.(12 votes)

- What is the name for this method?(2 votes)
- It's called "Rule of Sarrus". It only works with 3x3 matrices.(10 votes)

- Is there such a thing as a 3d matrix, and would we use this to find its determinant?(5 votes)
- Yes, you can have multidimensional matrices, but they're generally called multidimensional "arrays." Here's an example of a 3d matrix that computers may use to store the colors of an image: http://www.mathworks.com/help/matlab/math/ch_data_struct5.gif

Unfortunately, I can't answer your second question since I've never learned how to find determinants of matrices higher than two dimensions.

Off topic, but the largest multidimensional array I've seen in practical use is six dimensions: http://blog.digilentinc.com/wp-content/uploads/2014/08/6D-array.gif(4 votes)

- Can we only solve system of linear equations through matrices . Can't we solve system of quadratic ,cubic ,quartic or any degree equations through matrices?(3 votes)
`Actually we can solve quadratic, cubic equations, etc... by using matrices as well`

Here's a link about using matrices in solving a quadratic equation:

http://answers.yahoo.com/question/index?qid=20081123080813AAAsKHF(3 votes)

- Can we find the determinant of an asymmetrical matrix like 2x3, 3x5 etc.?(2 votes)
- No, only square (n x n) matrices have a determinant.(3 votes)

- Hello another great tutorial :) My question is: How to find the determinant of a 4x4 matrix? I really need this because I have exam on Monday !! :) Anyone?(3 votes)
- I believe you must use the other method found here: https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-determinants-and-inverses-of-large-matrices/v/finding-the-determinant-of-a-3x3-matrix-method-2

You can then use the method in THIS video to find the determinant of those 3x3 matrices(1 vote)

## Video transcript

As another hint, I will take the same matrix, matrix A and take its determinant again but I will do it using a different technique, either technique is valid so here we saying what is the determinant of the 3X3 Matrix A and we can is we can rewrite first two column so first column right over here we could rewrite it as 4 4 -2 and then the second column right over here we could rewrite it -1 5 0 and we could do is we could take the sum of the products of the first three top left bottom left diagonal, let me show you so the product of that that plus that plus that trying my best to draw this neatly and then from that subtract top right to bottom left diagonal so from that subtract let get a color I havent used subtract that and that and that sounds really confusing with all the diagonals i have drawn lets look at the blue ones first 4 times 5 times 0 4 times 5 times 0 plus -1 times 3 times -2 plus -1 times 3 times -2 let me put these in parenthesis plus 1 times 4 times 0 and then we gonna subtract all of these orange diagonal, we go from the top right to the bottom left so we gonna subtract we could do 1 times 5 times -2 1 times 5 times -2 and then we can subtract subtract 4 times 3 times 0 4 times 3 times 0 and then we can subtract -1 times 4 times 0 -1 times 4 times 0 and now we just evaluate this over here 4 times 5 times 0 is just 0 -1 times 3 times -2 is +6 so this is +6 1 times 4 times 0 is 0 once again and then we have 1 times 5 times -2 is -10 but we have this negative out here so it becomes a +10 then 4 times 3 times , well thats just going to be 0. and then we have -1 times 4 times 0, which is just 0 so we are left with +6 + 10 which is equal to +16