# How To Get The Inverse In A 3x 3 Matrix?

• Mar 22nd 2011, 11:10 AM
AlphaRock
How To Get The Inverse In A 3x 3 Matrix?
A = [3x3] Matrix =
-1 3 -1
2 -2 3
-1 1 2

How Do We Get (A)^(-1) (The inverse of A)?

• Mar 22nd 2011, 11:19 AM
Ackbeet
Try this: append the matrix $\displaystyle I$ to $\displaystyle A$ thus:

$\displaystyle [A|I]=\left[ \begin{array}{rrr|rrr} -1 &3 &-1 &1 &0 &0\\ 2 &-2 &3 &0 &1 &0\\ -1 &1 &2 &0 &0 &1 \end{array}\right].$

Now perform on this entire matrix the row operations necessary to get to reduced row echelon form; that is, you take $\displaystyle A\to I.$ Then you'll have

$\displaystyle [A|I]\to[I|A^{-1}].$

All of this assumes, of course, that $\displaystyle A$ is invertible.

Make sense?
• Mar 22nd 2011, 12:34 PM
pickslides
If Ackbeet's method is a bit tricky then try using some software like MATLAB to solve it for you.
• Mar 22nd 2011, 12:40 PM
dwsmith
Quote:

Originally Posted by AlphaRock
A = [3x3] Matrix =
-1 3 -1
2 -2 3
-1 1 2

How Do We Get (A)^(-1) (The inverse of A)?

$\displaystyle \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A$

Which is why singular matrices don't have an inverse.
• Mar 22nd 2011, 12:48 PM
pickslides
Quote:

Originally Posted by dwsmith
$\displaystyle \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A$

Hey dw, is this correct? It may be too early in the mornig for me but it has sent my head spinning.
• Mar 22nd 2011, 12:50 PM
Ackbeet
Quote:

Originally Posted by dwsmith
$\displaystyle \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A$

Which is why singular matrices don't have an inverse.

Technically, you need to have the transpose of the cofactor matrix there, not A. As pickslides pointed out.
• Mar 22nd 2011, 12:51 PM
dwsmith
Quote:

Originally Posted by Ackbeet
Technically, you need to have the transpose of the cofactor matrix there, not A. As pickslides pointed out.

I realized that when I read his post. 50% was right though!
• Mar 22nd 2011, 01:00 PM
Ackbeet
Quote:

Originally Posted by dwsmith
I realized that when I read his post. 50% was right though!

Right-ho.
• Mar 23rd 2011, 04:48 AM
mr fantastic
Quote:

Originally Posted by AlphaRock
A = [3x3] Matrix =
-1 3 -1
2 -2 3
-1 1 2

How Do We Get (A)^(-1) (The inverse of A)?