If A, B Are Two N × N Non-singular Matrices, Then - Mathematics

Question

If A, B are two n × n non-singular matrices, then __________ .

Options

AB is non-singular
AB is singular
\[\left( AB \right)^{- 1} A^{- 1} B^{- 1}\]
(AB)⁻¹ does not exist

MCQ

Solution

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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Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method

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Chapter 7: Adjoint and Inverse of a Matrix - Exercise 7.4 [Page 37]

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RD Sharma Mathematics [English] Class 12

Chapter 7 Adjoint and Inverse of a Matrix
Exercise 7.4 | Q 6 | Page 37

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