We study the link between the complexity of polynomial matrix multiplication and the complexity of solving other basic linear algebra problems on polynomial matrices. By polynomial matrices we mean n x n matrices of degree d over K[x] where K is a commutative field. Under the straight-line program model we show that multiplication is reducible to the problem of computing the coefficient of degree d of the determinant. Conversely, we propose algorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplication; for the determinant, the straight-line program we give also relies on matrix product over K[x] and provides an alternative to Storjohann's determinant algorithm. We further show that all these problems can be solved in particular in O~(n^w d) operations in K. Here the "soft Oh'' notation O~ indicates some missing log(nd) factors and w is the exponent of matrix multiplication overK.