Define matrices A =

 a 0 1 a



, B =

 b 1 0 b



.

Show that AB = BA and that eAeB = ea+b

 1 1 1 2 

eA+B = ea+b



cosh(1)  1 0 0 1 

+ sinh(1)  0 1 1 0  

.

Hence, eAeB = eA+B. (Hint: Use Problem 10 to calculate eA and eB.

For eA+B write A + B = (a + b)I + R with R satisfying R2 = I.)