Nonsingular, constant multiples of a fundamental matrix are fundamental

Let \frac{d}{dt} Y = AY be a linear system of first order differential equations over \mathbb{R} as in this previous exercise. Suppose M is a fundamental matrix of this system (i.e. a matrix whose columns are linearly independent solutions) and let Q be a nonsingular matrix over \mathbb{R}. Prove that MQ is also a fundamental matrix for this differential equation.


Write Q = [Q_1\ \ldots\ Q_n] as a column matrix. Now \frac{d}{dt} MQ = \frac{d}{dt} [MQ_1\ ldots\ MQ_n] = [(\frac{d}{dt}M)Q_1\ \ldots\ (\frac{d}{dt} M)Q_n] by the definition of our matrix derivative and part 2 of this previous exercise. This is then equal to (\frac{d}{dt} M)Q = AMQ. In particular, the columns of MQ are solutions of \frac{d}{dt} Y = AY. Moreover, since Q is nonsingular, MQ is nonsingular. So MQ is also a fundamental matrix.

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