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A new block-Routh array with block-Routh algorithm is developed to extract the greatest common matrix polynomial of two matrix polynomials that are not coprime, and to construct a block-transformation matrix that transforms a block-state equation from a block-companion form to a block-tridiagonal form. The newly developed block-state equation in the block-tridiagonal form is a minimal realisation of a matrix-transfer function. Also, the block-state equation is used to synthesise a driving-point impedance matrix. A stability criterion is then derived to test the stability of a class of matrix transfer functions.