The matrices$$\left[ {\matrix{ {\cos \,\theta } & { - \sin \,\theta } \cr {\sin \,\,\theta } & {\cos \,\,\theta } \cr } } \right]\,\,and$$
$$\left[ {\matrix{ a & 0 \cr 0 & b \cr } } \right]\,$$ commute under multiplication

The formula used to compute an approximation for the second derivative of a function $$f$$ at a point $${x_0}$$ is

Let $$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} \cr {{a_{21}}} & {{a_{22}}} \cr } } \right]\,\,$$ and $$B = \left[ {\matrix{ {{b_{11}}} & {{b_{12}}} \cr {{b_{21}}} & {{b_{22}}} \cr } } \right]\,\,$$ be
two matrices such that $$AB=1.$$
Let $$C = A\left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]$$ and $$CD=1.$$
Express the elements of $$D$$ in terms of the elements of $$B.$$

The process state transition diagram in Figure is representative of