# Matrix question.

• Jan 31st 2011, 04:23 PM
grooverandshaker
Matrix question.
If A is a 2X2 matrix, a12 = a21 = 0, a11 ≠ 0, a22 ≠ 0 then show that A is regular and find A^-1.
• Jan 31st 2011, 04:35 PM
topsquark
Quote:

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

A matrix is regular if it has an inverse. How can you tell if a matrix has an inverse? (Hint: Think of what the determinant of a singular matrix is.)

-Dan
• Jan 31st 2011, 04:41 PM
grooverandshaker
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 PM
rtblue
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 PM
topsquark
Quote:

Originally Posted by grooverandshaker
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?

$\displaystyle \displaystyle A=\begin{bmatrix}\alpha & 0\\0&\beta\end{bmatrix} \ \ \ \ \alpha, \ \beta\neq 0$
$\displaystyle \displaystyle\text{det}(A)=\alpha\cdot\beta-0\neq 0$
$\displaystyle \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A$