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Question
If A and B are non-singular matrices of the same order, write whether AB is singular or non-singular.
Solution
Let A & B be non-singular matrices of order n.
\[\left| A \right| \neq 0 \text{ and }\left| B \right| \neq 0 \left[\text{ By definition }\right] \]
Since they are of same order,
\[\left| AB \right| = \left| A \right|\left| B \right|\]
\[\left| AB \right| = 0\text{ iff either }\left| A \right| = 0\text{ or }\left| B \right| = 0 \]
\[\text{ But it is not the case here . Thus, }\left| AB \right|\text{ is non - zero and AB is non - singular matrix }.\]
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