If ##A## is ##m \times n## matrix, ##B## is an ##n \times m## matrix and ##n < m##. Then show that ##AB## is not invertible.(adsbygoogle = window.adsbygoogle || []).push({});

Let ##R## be the reduced echelon form of ##AB## and let ##AB## be invertible.

##I = P(AB)## where ##P## is some invertible matrix.

Since ##n < m## and ##B## is ##n \times m## therefore there is a non trivial solution to ##B\mathbf X = 0##

Let it be ##\bf X_0##

##I\mathbf X = P(AB)\mathbf X_0 \implies \mathbf X_0 = PA (B \mathbf X_0) = PA \times 0 = 0##

Which means ##\mathbf X_0 = 0##, which is contradiction. Therefore ##AB## is not invertible.

Is this correct ?

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# B Invertibility of a Matrix

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