| Assertion [A]: For two matrices A and B of order 3, \[\left| A \right|=3,\,\,\,\left| B \right|=-4\], then \[\left| 2AB \right|\] is \[-96\]. |
| Reason [R]: For a matrix A of order n and a scalar k, \[\left| kA \right|={{k}^{n}}\,\left| A \right|\]. |
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 :
Here, \[\left| 2AB \right|={{2}^{3}}\left| AB \right|=8\left| A \right|\,\,\,\left| B \right|\] \[=8\times 3\times -4=-96\] \[\therefore \]Assertion [A] is true \[\left\{ \because \,\,\left| kA \right|={{k}^{n}}\,\left| A \right|\,and\,\left| AB \right|=\left| A \right|\,\left| B \right| \right\}\] Also we know that \[\left| kA \right|={{k}^{n}}\,\left| A \right|\,\]for matrix A of order n. \[\therefore \]Reason (R) is true But \[\left| AB \right|=\left| A \right|\,\,\,\left| B \right|\]is not mentioned in Reason R. \[\therefore \]Both A and R are true but R is not correct explanation of A Hence option [B] is the correct answer.
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