smc

Taking the prime factorization of $161$ reveals that it is equal to $23\ast 7.$ Therefore, the only ways to represent $161$ as a product of two positive integers is $161\ast 1$ and $23\ast 7.$ Because neither $161$ nor $1$ is a two-digit number, we know that $a$ and $b$ are $23$ and $7.$ Because $23$ is a two-digit number, we know that a, with its two digits reversed, gives $23.$ Therefore, $a=32$ and $b=7.$ Multiplying our two correct values of $a$ and $b$ yields $$a\ast b=32\ast 7=$$ $$=\boxed{224}$$