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प्रश्न
If A, B are two n × n non-singular matrices, then __________ .
विकल्प
AB is non-singular
AB is singular
\[\left( AB \right)^{- 1} A^{- 1} B^{- 1}\]
(AB)−1 does not exist
उत्तर
AB is non-singular
A and B are non - singular matrices of order n × n.
\[ \therefore \left| A \right| \neq 0\text{ and }\left| B \right| \neq 0 . . . \left( 1 \right)\]
A and B are of the same order, so AB is defined and is of the same order .
Thus,
\[\left| AB \right| = \left| A \right|\left| B \right| \]
\[ \Rightarrow \left| AB \right| \neq 0 \left[\text{ Using }\left( 1 \right) \right] \]
Thus, AB is non - singular .
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