Hello arze Originally Posted by

**arze** so to find the matrices of a series of transformations:

A followed by C I'll get $\displaystyle \left(\begin{array}{cc}1&0\\0&-1\end{array}\right)\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)=\left(\begin{array}{cc}0&-1\\-1&0\end{array}\right)$?

and A following C would be $\displaystyle \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$

The answers for these two are reversed.

C carried out after E $\displaystyle \left(\begin{array}{cc}0&-1\\1&0\end{array}\right)\left(\begin{array}{cc}0&1 \\1&0\end{array}\right)=\left(\begin{array}{cc}1&0 \\0&-1\end{array}\right)$

answer is $\displaystyle \left(\begin{array}{cc}-1&0\\0&1\end{array}\right)$

and also A followed by B followed by C $\displaystyle \left(\begin{array}{cc}1&0\\0&-1\end{array}\right)\left(\begin{array}{cc}2&0\\0&2 \end{array}\right)\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)=\left(\begin{array}{cc}2& 0\\0&-2\end{array}\right)\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)=\left(\begin{array}{cc}0&-2\\-2&0\end{array}\right)$

answer is $\displaystyle \left(\begin{array}{cc}0&2\\2&0\end{array}\right)$

thanks

You need to remember that you work from *right to left *when you're combining transformation matrices. So to find the matrix that represents $\displaystyle A$ followed by $\displaystyle C$, you need to work out the product $\displaystyle CA$.

This is because the transformation matrix is always written on the *left *of the object that it's working on. So if I transform $\displaystyle (p,q)$ with matrix $\displaystyle A$, I'll get:$\displaystyle A\left(\begin{array}{c}p\\q\end{array}\right)$$\displaystyle =\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)\left(\begin{array}{c}p\\q\end{ array}\right)$

Then if we do $\displaystyle C$ to this, we get:$\displaystyle CA\left(\begin{array}{c}p\\q\end{array}\right)$$\displaystyle =\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)\left(\begin{array}{cc}1&0 \\0&-1\end{array}\right)\left(\begin{array}{c}p\\q\end{ array}\right)$

So $\displaystyle A$ followed by $\displaystyle C$ is $\displaystyle CA$, not $\displaystyle AC$.

Can you see where you've gone wrong now?

Grandad