Can you easily invert 3 by 3 matrices using a simple formula?

[The Divine Zephyr]

Anyone know how to do it? Please provide an easy explanation. Please help. Thank you.

 

[Muzza]

Isn't it like inverting any other form of matrix? You write up your matrix and the identity matrix like so:

a b c | 1 0 0
d e f | 0 1 0
g h i | 0 0 1

And perform Gaussian elimination until you reach:

1 0 0 | x y z
0 1 0 | p q w
0 0 1 | r t u

Then the matrix to the right of the |-signs is the inverse you're looking for.

 

[HallsofIvy]

Yep. There are other methods (cofactors divided by determinant) but row-reduction is the simplest.

 

[The Divine Zephyr]

Originally posted by Muzza
Isn't it like inverting any other form of matrix? You write up your matrix and the identity matrix like so:

a b c | 1 0 0
d e f | 0 1 0
g h i | 0 0 1

And perform Gaussian elimination until you reach:

1 0 0 | x y z
0 1 0 | p q w
0 0 1 | r t u

Then the matrix to the right of the |-signs is the inverse you're looking for.

Awesome! But.. umm... what is Gaussian elimination?

 

[HallsofIvy]

If you are going to ask questions about Linear Algebra it would be a good idea to read at least the first few chapters of a textbook on linear algebra!

"Gaussian Elimination" is basically the method of "elimination of variables" to solve a system of equations- multiply one equation by a number, add to another in order to eliminate one of the variables. It is often used specifically to denote the same thing applied to matrices of coefficients of the equations.

 

[The Divine Zephyr]

Oh, that? I get it. Thank you.

 

[franz32]

My reply

hello.

It would also be better to practice solving for the inverse of a matrix. And, it would be easier for you if you know know the topic by heart. =)

Of course, not all matrices have an inverse, or what we call a nonsingular matrix. This is very "special" later in the topic.
One example is the use of determinants.

Start with basics

 

[phantomAI]

Originally posted by franz32

Of course, not all matrices have an inverse, or what we call a nonsingular matrix. This is very "special" later in the topic.
One example is the use of determinants.

Start with basics

Yeah if the determinant of a matrix is zero, it does not contain an inverse and is a nonsingular matrix.

 

[selfAdjoint]

You've got singular and nonsingular backward. Singular matrices have zero determinants and don't have inverses, because in computing the inverse you divide by the determinant (even if you don't think you do!) and dividing by zero is a "singular" mathematical operation, i.e. not defined.

Nonsingular matrices do have inverses, and necessarily then they have nonzero determinants.

 

[Singularity]

I believe the formula for the inverse of a nxn matrix is

inverse of A = 1/(det(A)) * adj(A)

i speak under correction. we did this last year and i have forgotten most of last year's stuff during the holidays