How Do I Calculate the Inverse of Matrix P and Determine Eigenvectors?

[matrix_204]

Could someone tell me how to get the P(inverse), P^-1. For example I read all the examples in my book and it has like given the matrix for P and then it finds the matrix for P(inverse)Vo , how do i find P^-1. Plz help me quickly.
Ex. P= | 2 -1 |
asdfasf| 3 as1 |
and Vo=| 1 |
iiiiiiiiiiiiiiiiii| 1 |

so P^-1Vo=1/5 [ 2 -1](transpose)

 

[cyby]

For two by two matrices, it is easy.

For a 2x2 matrix

$$A = \left(\begin{array}{cc}a&b\\c&d\end{array}\right)$$

The inverse is just...

$$A^{-1} = \frac{1}{\|A\|} \left(\begin{array}{cc}d&-b\\-c&a\end{array}\right)$$

 

[matrix_204]

thank you very much, it saved me so much time, also, is there a formula for a 3x3 matrix too or no.

btw is ||A||= a^2 + b^2 - c^2 - d^2,
just wondering, i don't know if that's right but what would it be for a 2x2 matrix.

 

[cyby]

To save me from typing up too much LaTeX code...

http://mathworld.wolfram.com/MatrixInverse.html

 

[matrix_204]

ok got it, thanx

 

[cyby]

And no, what I had

$$\|A\|$$ is the determinant of A. For a 2x2 matrix

$$A = \left(\begin{array}{cc}a&b\\c&d\end{array}\right)$$

$$\|A\| = ad - bc$$

 

[matrix_204]

I have one more question, because sometimes its hard to figure out the eigenvector, i was wondering is there a easier way to figure what what the eigenvectors are besides using the equation
http://mathworld.wolfram.com/eimg422.gif

 

[cyby]

I don't recall so. But please, do yourself a favor and don't work them out by hand. It's just too much boring arithmetic...

 

[shmoe]

I have one more question, because sometimes its hard to figure out the eigenvector, i was wondering is there a easier way to figure what what the eigenvectors are besides using the equation
http://mathworld.wolfram.com/eimg422.gif

Just to clarify terminology, that equation you linked to gives the eigenvalues, which you then use to find the eigenvectors by looking at the nullspace of $$A-\lambda I$$, where $$\lambda$$ is an eigenvalue.

I do suggest you work these out by hand when first learning them. You're more likely to understand what an eigenvector is if you're swimming through the arithmetic trenches than if you're simply entering a matrix into a computer or calculator and having it spit out some answers for you. Of course if you feel you have fully mastered the concept, by all means use mechanical aid (and certainly don't shy from using it to check your work). Just my opinion.