Let A be a \(3\times3\) matrix such that \(A^2-5A+7I=O\).
Statement - I : \(A^{-1}=\frac{1}{7}(5I-A)\).
Statement - II : The polynomial \(A^3-2A^2-3A+I\) can be reduced to \(5(A-4I)\).
Then :
Statement - I : \(A^{-1}=\frac{1}{7}(5I-A)\).
Statement - II : The polynomial \(A^3-2A^2-3A+I\) can be reduced to \(5(A-4I)\).
Then :
Show Hint
- Statement-I is true, but Statement-II is false.
- Statement-I is false, but Statement-II is true.
- Both the statements are true.
- Both the statements are false.
The Correct Option is C
Solution and Explanation
Statement II: \(A^2=5A-7I\), so \(A^3=5A^2-7A=5(5A-7I)-7A=18A-35I\).
\(A^3-2A^2-3A+I=(18A-35I)-2(5A-7I)-3A+I=5A-20I=5(A-4I)\). True.
Both statements are true.
Answer: (3) Both the statements are true.
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