If A is a 2X2 matrix, a12 = a21 = 0, a11 ≠ 0, a22 ≠ 0 then show that A is regular and find A^-1.

Many thanks in advance.

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- Jan 31st 2011, 04:23 PMgrooverandshakerMatrix question.
If A is a 2X2 matrix, a12 = a21 = 0, a11 ≠ 0, a22 ≠ 0 then show that A is regular and find A^-1.

Many thanks in advance. - Jan 31st 2011, 04:35 PMtopsquark
- Jan 31st 2011, 04:41 PMgrooverandshaker
Um, I know that a matrix has an inverse if the determinant is nonzero. Is that right?

I usually know how to find the determinant and the inverse but I just can't do it when the matrix isn't written out.

Could you help me figure out what the original matrix is? - Jan 31st 2011, 05:00 PMrtblue
Given that a12 and a21 are both 0 and that a11 and a22 are not 0, the matrix must have a nonzero determinant. There are no values of a11 and a22 that can make the determinant 0.

- Jan 31st 2011, 05:13 PMtopsquark

This page seems to be a good guide in general. It has an explicit answer for the 2 x 2 case, but make sure you understand how to do the problem in general.

-Dan - Jan 31st 2011, 06:11 PMdwsmith
$\displaystyle \displaystyle A=\begin{bmatrix}\alpha & 0\\0&\beta\end{bmatrix} \ \ \ \ \alpha, \ \beta\neq 0$

$\displaystyle \displaystyle\text{det}(A)=\alpha\cdot\beta-0\neq 0$

$\displaystyle \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A $ - Jan 31st 2011, 08:53 PMgrooverandshaker
Thanks guys. I think I understand now. :D