Thread: How To Get The Inverse In A 3x 3 Matrix?

1. 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)?

I would show an attempt at the solution, however, I am clueless about this.

2. 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?

3. If Ackbeet's method is a bit tricky then try using some software like MATLAB to solve it for you.

4. 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)?

I would show an attempt at the solution, however, I am clueless about this.
$\displaystyle \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A$

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

5. 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.

6. 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.

7. 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!

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

9. 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)?

I would show an attempt at the solution, however, I am clueless about this.
Don't you have class notes or a textbook that explain what to do? (And if you don't, why are you attempting this question?)