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### Course: Algebra (all content)>Unit 20

Lesson 15: Determinants & inverses of large matrices

# Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix

Sal shows how to find the inverse of a 3x3 matrix using its determinant. In Part 1 we learn how to find the matrix of minors of a 3x3 matrix and its cofactor matrix. Created by Sal Khan.

## Want to join the conversation?

• I didnt quite understand the end of the video, at . I got the cofactor Matrix, but then what's left to do to get to the inverse of matrix C? Multiply the cofactor Matrix by which determinant, the one from C or the one from the cofactor Matrix?
• Don't listen to sal at the end of part 1 your supposed to find the TRANSPOSE of the co-factor matrix. Then multiply the transpose of the co-factor matrix by the determinant of the original matrix. Then you have the inverse.
• What are applications for this in the world?
• If you have ever been given a problem with three linear equations in three unknowns (x,y and z), and you worked through a big tedious mess of solving the system of equations by gathering like terms, expressing one variable in terms of the others, and substituting, etc, then you have already been doing this math!

Working with matrices is a way of turning the methods you already know into an automated, orderly system.

The real- world applications are any situation where there are three variables that relate to each other. The practical applications of setting up the math with matrices is that the steps can be programmed to be run by a computer.
• can determinant be found only for a square matrix?
• Yes. Determinant can only be found in square matrices like 2x2, 3x3, ....and so on.
• I'm just wondering, is it possible to find the determinant of a 1 by 1 matrix?
• Does anyone else skip the whole "checkerboard" thing and just use determinants that are the correct signs to begin with? If you just imagine the original matrix repeats infinitely in every direction, you can always go to the bottom-right of the number and grab that determinant square. So, for the `-2` entry, you'd go down and to the right and grab the square of:
``1 25 3``
and take the determinant of that, which is `-7`. This immediately gets you the `-7` in the cofactor matrix Sal writes at in the end, but avoids having to remember to do with checkerboard later. Any downside to that method?
• I quite like that approach. It makes it less likely to mess up the deletion of the row and column to get the minor, and the sign gets taken care of by the reversals, as appropriate, of the original two rows and columns. But, as Sal said, it's still not an operation I look forward to carrying out.
• Can you show an example of solving a 3x3 matrix solving for an X,Y,Z linear equation? I'm trying to work one out for the first time, I found the determinant, and the inverse, multiplied the inverse by the constants, and then multiplied that result by 1 over the determinant, my answer came out all messed up. I'm enjoying this, but it's frustrating when the substitution method is working better for me, and I want to be better than that.
• I think it's important to get a 'bigger picture' of why we use the inverse of the matrix to solve systems of linear equations. I'm guessing you're familiar with a system of equations like
``1x + 2y+3z = 52x + 3y + 1z = 63x + 7y + 2z = 8``

This is written in matrix form:
`A*x = b`, where `x` in this example is a vector of variables `[x ; y ; z]`. To solve for `x`, we premultiply both sides of the equation by the inverse of `A`:
`inv(A)*A*x = inv(A)*b`, and since `inv(A)*A = I`, the identity matrix,
`x = inv(A)*b`.

From your description, it looks like you accidentally multiplied by 1/det(A) when it wasn't necessary. The determinant is only used to find the inverse itself.

However, finding the inverse is (as you found out first hand), pretty difficult and prone to error. So people have worked out ways of solving the same problem `A*x=b` using other methods, one of which is using what is called LU decomposition. Going into that will be another big can of worms, so if you're interested you can look into that online.
• Unless I missed it, Sal never told us we first need to take the Transpose of the original matrix before doing the matrix of minors and multiplying by 1 over the Determinant. Is that covered in another video?
• Matrices seem tailor-made to be solved by computers... But is there a place where matrices are used outside of computers?
(i.e. a situation where I would have to solve if my computer broke)
• which method is used here? the one of cramer or gauss jordan?
(1 vote)
• its definitly not gauss jordan. @ Gauss-Jordan you transform your matrix to be the identity matrix. 1. you write both matrix and the identity matrix side by side. So what you see is like a 3x6 matrix (first three columns are the matrix and second 3 columns are the identity) 2.Now you use simple operations on them to get the identity matrix on your left 3 columns, if you have done this, then the right 3 columns are now the inverse of your matrix. Hopefully its not too confusing. But thats the way how to build the inverse by gauss-jordan. This method that sal uses is the conclusion of the cramers rule I think.