Question

$\text{Statement 1: If}{B}^{-1}=\left[\begin{array}{ccc}0& 6& 7\\ 8& 9& 10\\ 11& 12& 13\end{array}\right]\text{and}A=\left[\begin{array}{ccc}5& 6& 7\\ 8& 9& 10\\ 11& 12& 13\end{array}\right],\text{then}(AB{)}^{-1}\text{does not exist}\text{Statement 2 : Since}|A|=0,(AB{)}^{-1}=B^{-1}{A}^{-1}\text{is meaningless.}$

• A. Both statement 1 and statement 2 are true and statement 2 is the correct explaination of statement 1.
• B. Both statement 1 and statement 2 are true but statement 2 is not the correct explaination of statement 1.
• C. Statement 1 is true but statement 2 is false.
• D. Statement 2 is true but statement 1 is false.

Answer 1

The correct option is A Both statement 1 and statement 2 are true and statement 2 is the correct explaination of statement 1.

$\text{Statement 1}:|A|=\left|\begin{array}{ccc}5& 6& 7\\ 8& 9& 10\\ 11& 12& 13\end{array}\right|$
Perfoming row operations as follows: $R_3\to R_3-R_2$
$|A|=\left|\begin{array}{ccc}5& 6& 7\\ 8& 9& 10\\ 3& 3& 3\end{array}\right|$
Now $R_2\to R_2-R_1$
$|A|=\left|\begin{array}{ccc}5& 6& 7\\ 3& 3& 3\\ 3& 3& 3\end{array}\right|=0$
So $A^{-1}$ doesn't exist.

$\text{We know that}(AB{)}^{-1}=B^{-1}{A}^{-1}\text{As}{A}^{-1}\text{doesn't exists,}(AB{)}^{-1}\text{doesn't exists.}\text{Statement 2 is also true}$

Hence both statements are true and statement 2 is correct explanation of statement 1.