Question

Consider the following statements
I. A rectangular matrix possesses inverse matrix.
II. If $B$ is the inverse of $A$, then $A$ is also the inverse of $B$.
III. Inverse of a square matrix, if it exists, is not unique.
IV. If $A$ and $B$ are invertible matrices of the same order, then $(AB{)}^{-1}=B^{-1}{A}^{-1}$.
Choose the correct option.

• A. I and III are true
• B. II and IV are true
• C. I and II are true
• D. III and IV are true

Answer 1

Answer: (B)

A rectangular matrix does not possess inverse matrix, since for products $BA$ and $AB$ to be defined and to be equal, it is necessary that matrices $A$ and $B$ should be square matrices of the same order.
If $B$ is the inverse of $A$, then $A$ is also the inverse of $B$.
Uniqueness of inverse Inverse of a square matrix, if it exists, is unique.
Let $A=\left[a_{ij}\right]$ be a square matrix of order $m$. If possible, let $B$ and $C$ be two inverses of $A$. We shall show that $B=C$.
Since, $B$ is the inverse of $A$.
$\Rightarrow AB=BA=1$ ...(i)
Since, $C$ is also the inverse of $A$
$\Rightarrow AC=CA=1$...(ii)
Thus, $B=BI=B(AC)=(BA)C=IC=C$
From the definition of inverse of a matrix, we have
$(AB)(AB{)}^{-1}=1$
or $A^{-1}(AB)(AB{)}^{-1}=A^{-1}$ I
(multiplying both sides by $A^{-1}$ )
or $\left(A^{-1}A\right)B(AB{)}^{-1}=A^{-1}\left(\because A^{-1}I=A^{-1}\right)$
or $IB(AB{)}^{-1}=A^{-1}$
or $B(AB{)}^{-1}=A^{-1}$
or $B^{-1}B(AB{)}^{-1}=B^{-1}{A}^{-1}$
or $I(AB{)}^{-1}=B^{-1}{A}^{-1}$
Hence,
$(AB{)}^{-1}=B^{-1}{A}^{-1}$