We study the structure of isometries defined on the algebra A of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry A→ M_n must be of the form either A UAU^* or A U AU^*, where A is the complex conjugation and U is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry A→ M_n(C) is of the form A UAV where U and V are two unitary matrices. This implies, in particular,

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... at every such an isometry is a complete isometry and that a unital linear isometry A→ M_n(C) is necessarily an algebra homomorphism.