If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Course: Algebra (all content)>Unit 20

Lesson 15: Determinants & inverses of large matrices

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?
• 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.
• Determinants are valid only for square matrices right?
• Yes, you can only calculate the determinant for a square matrix.
• 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.
• Yes you can! Actually using that row would be a handy trick to calculate the determinant a little faster.
• what is a checker board pattern? Sal mentions it at
• 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
• So for 4x4 matrix, checker board patern would be + - + - and for 5x5 it would be + - + - +?
• 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
• Why does he say "AS A HINT"?
• Yeah, I thought so too. I found a nice intro here, everyone should watch this first.
• 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??
• 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 cd e fg 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.
• 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.
• 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). ^_^