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One of the simplest matrix-valued function with a single variable matrix $X$is given by $A + BXC$. In this this note, analytical formulas are establishedfor calculating the maximal and minimal ranks of $A + BXC$ when the rank of thevariable matrix $X$ is fixed by using a simultaneous decomposition of $A$, $B$and $C$ and some preliminary results. Some applications of the formulas incompleting partially-specified block matrix with the maximal and minimal ranksare also given.

Analytical formulas for calculating the extremal ranks of the matrix-valued function $A + BXC$ when the rank of $X$ is fixed