| Assertion [A]: adj A is a non-singular matrix. |
| Reason [R]: A is non singular matrix. |
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: A
Solution :
| Given Assertion adj A is non singular |
| \[\Rightarrow \,\,\left| adj\,\,A \right|\ne 0\] |
| We know that \[A\,\,adj\,\,\,A=\,\,\left| A \right|\,\,{{I}_{n}}\] |
| \[\Rightarrow \,\,\left| A\,\,adj\,\,A \right|={{\left| A \right|}^{n}}\,\,\left| {{I}_{n}} \right|\] |
| \[\Rightarrow \,\,\left| A \right|\,\,\left| adj\,\,A \right|={{\left| A \right|}^{n}}\,\,.\,\,1\] |
| \[\Rightarrow \,\,\,\,\left| adj\,\,A \right|={{\left| A \right|}^{n-1}}\] |
| But \[\left| adj\,\,A \right|\ne 0\Rightarrow {{\left| A \right|}^{n-1}}\,\ne 0\Rightarrow \left| A \right|\ne 0\] |
| \[\Rightarrow \]Matrix A is non singular \[\Rightarrow \]R is true. |
| \[\therefore \]Both A and R are true and R is the correct explanation of A |
| Hence option [A] is the correct answer. |
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