Linear algebra problem!!!!

if A is singular , then all splittings A = S - T must fail. From Ax = 0, show that S-1Tx = x. So this matrix B = S-1T has ? = 1 and fails.

A = S-T

Ax = 0 ==> (S-T)x = 0 ==> Sx = Tx ==> x = S^(-1)T x ==> x = Bx

thus, B = S^(-1)T has eigenvalue 1.