Let $A$ be a square matrix of order 3 . Choose the correct option regarding the following statements I.…
Let $A$ be a square matrix of order 3 . Choose the correct option regarding the following statements
I. There exists a matrix $B$ of order 3 such that $A B=I_3$
II. There exists a matrix $C$ of order 3 such that $C A=I_3$
III. $A$ is invertible
Only III implies I and II
I, II and III are equivalent statements
In I and II, B can be different from C
None of the above
Solution
Since, given statement (I) $A B=I_3$
$
\Rightarrow \quad B=I_3 A^{-1} \Rightarrow B=A^{-1}
$
Statement (II) $C A=I_3 \Rightarrow C=I_3 \times A^{-1}$
$
C=A^{-1}
$
Statement(II) From Eqs. (i) and (ii), $A$ is invertable matrix
Given Statements (I), (II), (III) are equivalent Hence, option (2) is correct