Question

A and b are two symmetric matrix, give an example to show that the products ab is not symmetric

Answer 1

Answer:

An example is

$\displaystyle\left(\begin{array}{cc}1&0\\0&0\end{array}\right)\left(\begin{array}{cc}2&1\\1&3\end{array}\right)=\left(\begin{array}{cc}2&1\\0&0\end{array}\right)$

Step-by-step explanation:

In general...

$\displaystyle\left(\begin{array}{cc}a&b\\b&c\end{array}\right)\left(\begin{array}{cc}d&e\\e&f\end{array}\right)=\left(\begin{array}{cc}ad+be&ae+bf\\bd+ce&be+cf\end{array}\right)$

So to get an example where the product is not symmetric, just need to choose numbers so that ae+bf ≠ bd + ce.

One example would be a = e = 1, b = c = 0.  Then ae+bf = 1 while bd+ce = 0.

For d and f we can take whatever we like.  For instance, let's take d = 2 and f = 3.  This gives the example above.