Hello everyone,

In my textbook I am asked to calculate the inverse matrix of $\displaystyle \begin{pmatrix}2 & -1\\1

& 3

\end{pmatrix}$ by a system of equations.

This is no problem at all, I quickly found that a = 3/7, b = 1/7, c = -1/7, d = 2/7 is the fitting set of solutions. It is then quickly proven that this inverse matrix can be written as $\displaystyle 1/7\begin{pmatrix}3 & 1\\-1

& 2

\end{pmatrix}$.

What I'm stuck with is proving that the inverse matrix for any matrix $\displaystyle \begin{pmatrix}p & q\\r

& s

\end{pmatrix}$ equals $\displaystyle 1/(ps-qr)\begin{pmatrix}s & -q\\-r

& p

\end{pmatrix}$

If I create the system:

ap + br = 1

cp + dr = 0

aq + bs = 0

cq + ds = 1

I am stuck with the variables a, b, c and d. Has anyone got a clue how to do this proof?

Note: Please excuse me for the random <br/> in the matrices, I am not yet very good at LaTeX.