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## Linear algebra

### Unit 2: Lesson 5

Finding inverses and determinants

# 3 x 3 determinant

Determinants: Finding the determinant of a 3x3 matrix. Created by Sal Khan.

## Want to join the conversation?

• Would the diagonal method work too? Because i think thats a lot simpler than this.
• The Sarrus rule only applies to 2x2 and 3x3 matrices. So it is better to learn this one-since it it universal! :D
• how do you take a determinant of a 2x3 matrix?
• Its not possible to find determinant of 2x3 matrix.Determinant can be done only for square matrix where dimension of row and column must be same.Like 3x3 or 4x4 matrices.Hope you got your answer.
• Maybe this is a silly question, but why do we care if c is invertible? Is there some other place we use this?

edit: (Thanks to newbarker for a good answer)
• If you were solving a system of 3 equations in 3 unknowns and wanted to know if there was a unique solution, then invertibility is essential to there being a unique solution.

Another example from computer graphics: I might have an object. Examples being a cube, or a teapot, or a human body, etc. The object is represented as a collection of 3D position vectors (that is vectors in R3). I might want to perform some kind of transformation on this object (e.g. rotate, stretch, flip) and then have the ability to untransform it to its original form later. The transformation* would be represented by a 3x3 matrix. This transformation when multipled by the position vectors that represent the object yields transformed position vectors, Now when I want to untransform it, I find the inverse of the transformation matrix, multiply it by the transformed position vectors, and the original vectors are provided. This is used massively in games and CAD.

* Without wanting to complicate things yet, I just want to add that using a 3x3 matrix means you cannot represent a translation (shift in position) in 3D. To do that using matrix multiplication, you need a 4x4 matrix and a 4D point. Ignore this bit for now if it doesn't make sense.
• Hi. This is the first time I have delved into matrices and I find them fascinating if a bit abstract. Question: When finding the determinant of a 3X3 matrix, it seems that we only use the top row to set up the computation. This, for me, appears counter-intuitive. Why don't we have to make the same manipulations and computations for all three rows? Are the other rows somehow incorporated when we set up the solution using the first row?
• The determinant of the matrix A is the same as the determinant of the transpose of A, thus you can use any row or column to find the determinant. The more zeros in a row or column the more preferable it is to use that row or column.
• For the people confused about the "chess board pattern", you can use the following rule to determine the sign: -1 ^ ( i + j ) where i = row and j = column.

So expansion over the 1st row (a11, a12, a13) results in a +, -, + pattern
a11 = -1 ^ (1+1) = 1 (positive: +)
a12 = -1 ^ (1+2) = -1 (negative: -)
a13 = -1 ^ (1+3) = 1 (positive: +).
• so in other words :
if sum of indicies even, then add (+)
if sum of indicies is odd, then sub.(-)
• Isn't this similar, or exactly the same, as finding the Cross Product of three 3-dimensional vectors? Thanks.
• The entries of the vector obtained from taking the cross product are given by taking determinants, however the determinant is very different from cross product in an important way: cross product is an operation between two vectors witch spits out a third (orthogonal) vector; whereas determinants operate on matrices and spit out scalar (numbers).
• I am just curious how someone came up with 3 x 3 determinant definition. 2 x 2 determinant makes sense, but not sure how that applies to 3 x 3
• So, what's the inverse of the 3x3 matrix?
(1 vote)
• A good way to invert a 3x3 matrix is to augment it with the identity matrix and then row reduce the left hand side while doing the operations to the augmented side.