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We establish the formulas of the maximal rank of a 3×3 partial banded block matrix where X; Y; and Z are three variant quaternion matrices subject to linear matrix equationsA1X = C1; XB1 = C2; A2Y = D1; YB2 = D2; A3Z = E1; ZB3 = E2: In order to demonstrate the feasibility of the result obtained, we present a necessary and sufficient condition for the solvability to the cubic system A1X = C1; XB1 = C2; A2Y = D1; YB2 = D2; A3Z = E1; ZB3 = E2;XYZ = J over the quaternion algebra.

The maximal rank of a kind of partial banded block matrix subject to linear equations