If a linear eq. System has only a single solution, the matrix has always an inverse?

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If i have a 3x3 matrix of a linear eq. System and i archieve an upper triangular matrix (means unique solution) does this form always have inverse matrix?

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a linear eq system has only a single solution (Cramer's system) iff the matrix associated to the system is squar (nombers variable = nombers equation) and the matrice is inversible (determinant not 0).