In 1978, Frost and Storey asserted that any bivariate polynomial matrix is equivalent to its Smith normal form if and only if the reduced minors of each order of the matrix generate the unit ideal. In this talk, we first demonstrate by constructing an example that for any given positive integer $s$ with $s \geq 2$, there exists a square bivariate polynomial matrix $M$ over $K[x,y]$ with the degree of $\det(M)$ in $y$ being equal to $s$, for which the condition that the reduced minors of each order of $M$ generate the unit ideal in $K[x,y]$ is not a sufficient condition for $M$ to be equivalent to its Smith normal form. Subsequently, we prove that for any square bivariate polynomial matrix $M$ where the degree of $\det(M)$ in $y$ is at most $1$, Frost and Storey's assertion holds. Using the Quillen-Suslin Theorem, we further extend our consideration of $M$ to rank-deficient and non-square cases.