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 $(A B)^{-1}=B^{-1} A^{-1}$.
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 $B A$ and $A B$ 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 $A=\left[a_{i j}\right]$ 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 $A$.
$\Rightarrow A B=B A=1$ ...(i)
Since, $C$ is also the inverse of $A$
$\Rightarrow A C=C A=1$...(ii)
Thus, $ B=B I=B(A C)=(B A) C=I C=C$
From the definition of inverse of a matrix, we have
$(A B)(A B)^{-1}=1$
or $ A^{-1}(A B)(A B)^{-1}=A^{-1}$ I
(multiplying both sides by $A^{-1}$ )
or $\left(A^{-1} A\right) B(A B)^{-1}=A^{-1} \left(\because A^{-1} I=A^{-1}\right)$
or $ I B(A B)^{-1}=A^{-1}$
or $ B(A B)^{-1}=A^{-1}$
or $ B^{-1} B(A B)^{-1}=B^{-1} A^{-1}$
or $ I(A B)^{-1}=B^{-1} A^{-1}$
Hence,
$(A B)^{-1}=B^{-1} A^{-1}$