Let \(A(t)=(a_{ij}(t))\) be an \(n\times n\) matrix of functions. If \(W=\{A(t)|\) \(A(x)A(y)=A(y)A(x)\), \(\forall x,y\in R\}\) then W is called the class of autocommuting matrices of functions. A system of linear differential equations with an autocommuting coefficient matrix is integrable. The author studies a simplification problem of the class W and its applications. The obtained results include and extend the correspondng ones of C. R. Johnson [Interlacing polynomials, Proc. Am. Math. Soc. 100, 401-404 (1987; Zbl 0622.15018)].