A new block bialternate sum of a partitioned square matrix with itself is defined and some basic properties established. It has similar properties to the bialternate sum, but it preserves the block structure of the summand. It is used in place of the block Kronecker sum and the block Lyapunov sum to solve maximal stability problems of stable systems under low-gain integral control and of singularly perturbed systems. It is also used to find the ’critical gains’ for a first-order controller with a scalar plant from which all stabilizing first-order controllers can be constructed. In each stability problem the internal dimensions of the resulting closed formulae are lower than those currently available.