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The controllability (observability) of a linear, time-invariant system is examined by reducing the system equations into their canonical form or by using the controllability (observability) matrix. To reduce into the canonical form requires row as well as column operations and consequently the original variables are not preserved. The controllability. (observability) matrix gives little information about the system as compared to its size and its construction. In this paper the controllability (observability) of a system is examined by the use of algebraic equations contained in the zero-state Laplace transform of state equations and an algorithm which only requires elementary row operations on real matrices is given to derive these algebraic equations. Besides providing an explicit explanation of controllability (observability) and displaying the structural properties of a system, these algebraic equations give a transformation which requires no matrix inversion and which leaves the controllable (observable) states invariant when reducing the system into its canonical form. The algorithm is still valid when the coefficient matrix of is not the identity matrix in the state equations. Therefore, in order to apply the algorithm matrix inversion is not needed when a change of variables takes place.