# [SOLVED] Matrices

• Mar 21st 2010, 04:38 AM
NewtoMath
[SOLVED] Matrices
A is a 2x2 matrix, a12=a21=0, a11 does not equal 0, a22 does not equal 0.

Show that A is regular and find inverse of A
• Mar 21st 2010, 11:19 AM
s_ingram
matrices
Hi NewtoMath,

To find the inverse of a (2x2) matrix, interchange the elements in the leading diagonal, change the signs of the elements in the other diagonal and divide by the determinant of the matrix.

We have $A = \left( \begin{array}{cc} a_{11} & 0 \\ 0 & a_{22} \end{array} \right)$

$A^{-1} = \frac{1}{det A} \left( \begin{array}{cc} a_{22} & 0 \\ 0 & a_{11} \end{array} \right)$

$det A = \frac{1}{a{11}a_{22}}$

So, $A^{-1} = \left( \begin{array}{cc} \frac{1}{a_{11}} & 0 \\ 0 & \frac{1}{a_{22}} \end{array} \right)$

clearly $AA^{-1} = I$ I tried to find out what a regular matrix was. There are various definitions to do with stochastic matrices. The most suitable was that a regular matrix is one for which AB = BA, which is true in this case.
• Mar 21st 2010, 09:01 PM
NewtoMath
Hey thanks ingram! can't believe I didn't figure what "a12=a21=0" meant (Dull)

In my book it means to be a regular square matrix when there is an existing inverse or determinent, like when it doesn't equal zero in which case it's a singular.

Thanks again.