| Assertion [A]: For a square matrix of order 2 \[{{A}^{-1}}=\frac{1}{5}\,adj\,A,\]so \[\left| 2A \right|=20\]. |
| Reason [R]: For a square matrix of order n, \[{{A}^{-1}}=\frac{1}{\left| A \right|}\,adj\,A\] and \[\left| adj\,\,A \right|={{\left| A \right|}^{n-1}}\]. |
A) Both A and R are individually true and R is the correct explanation of A.
B) Both A and R are individually true and R is not the correct explanation of A.
C) 'A' is true but 'R' is false
D) 'A' is false but 'R' is true
E) Both A and R are false.
Correct Answer: B
Solution :
Given \[{{A}^{-1}}=\frac{1}{5}adj\,A\] We know that \[{{A}^{-1}}=\frac{1}{\left| A \right|}\,adj\,A\] \[\therefore \,\,\left| A \right|=5\] Also \[\left| 2A \right|={{2}^{2}}\,\,\left| A \right|\] \[\left\{ \because \,\,\left| kA \right|={{k}^{n}}\,\left| A \right| \right\}\] \[=4\times 5=20\] \[\therefore \]Given Assertion [A] is true. Also Reason (R) is true Since \[\left| kA \right|={{k}^{n}}\,\left| A \right|\]is not mentioned in Reason \[\therefore \]Reason (R) is not correct explanation of A Hence option [B] is the correct answer.
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