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

Inverting a 3x3 matrix using Gaussian elimination

Sal explains how we can find the inverse of a 3x3 matrix using Gaussian elimination. Created by Sal Khan.

Want to join the conversation?

• Okay, I maybe going to far with this question, but can you do ANYTHING to one side of the matrix to get it to be the identity matrix as long as you do it to the other side and still get the inverse?
• Nope, not anything. You must preserve row equivalence, which in practice means you can only use the three operations stated in the video: (1) interchange two rows, (2) multiply the elements of a row by a number different than 0 and (3) adding the elements of a row to the corresponding elements of another row.
• What is the identity matrix of a non-square matrix?
• Gauss-Jordan elimination is a lot faster but only for certain matrices--if the inverse matrix ends up having loads of fractions in it, then it's too hard to see the next step for Gauss-Jordan and the determinant/adjugate method is the only way I can solve the problem without pulling my hair out.
My question is: Is there an easy way to be able to tell, before you start the problem, if the matrix is simple and non-fraction-y enough that Gauss-Jordan is a reasonable tool, or if I should just go straight to finding the cofactor, determinant, and adjugate?
• I would say just go for it. Try the row elimination and then go to the longer method if this gets too hairy. If not, well then you just solved your matrix inverse and you can now apply it in whatever you're working on.
(1 vote)
• When doing row operations on an matrix, does that change the determinant? and if so is that amount predictable?
• It depends on the sort of row operation you are using. The three main row operations will do the following to the determinant:
Ri -> Ri + n*Rj (adding a row multiplied by n to another row)
This sort of row operation will not change the determinant.
Ri -> n*Ri (multiplying a row with n)
This one will change the determinant. The new determinant will be equal to the original determinant multiplied by n.
Ri <-> Rj (swap two rows)
This one will also change the determinant. The sign of the determinant will be opposite.
• So, would the Gauss-Jordan elimination method work for all square matrices?
• how do you find the inverse of the identity matrix?
• If you know how to invert matrices, you can use the same method for the identity matrix.

You will find that the inverse of the identity matrix is the identity matrix.
• How do you know to add/subtract vs. multiply can you just do whatever you want to as long as you get the identity or as long as you times one row by something then add it to another. Can you do that multiple times to get the identity? There seems to be many ways to do this unless I am not understanding it correctly...