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  • Let A be a 3 × 3 matrix such that A2 − 5A + 7I = O. Statement - I : <img alt="A^{-1}\, =\frac{1}{7}\left ( 5I\, -\, A \right )." src="https://learn.careers360.com/l

Let A be a 3 × 3 matrix such that A2 − 5A + 7I = O.

Statement - I : A^{-1}\, =\frac{1}{7}\left ( 5I\, -\, A \right ).

Statement - II : The polynomial A3 − 2A2 − 3A + I can be reduced to 5(A − 4I).

Option: 1

Statement-I is true, but Statement-II is false.


Option: 2

Statement-I is false, but Statement-II is true.


Option: 3

Both the statements are true.


Option: 4

Both the statements are false.


Answers (1)

best_answer

Given that

\\\mathrm{A^2-5A+7I=O} \\\Rightarrow \mathrm{A^2=5A-7I} \\\Rightarrow \mathrm{AA=5A-7I} \\\Rightarrow \mathrm{AAA^{-1}=5AA^{-1}-7IA^{-1}}\\\Rightarrow \mathrm{AI=5I-7A^{-1}}\\\Rightarrow \mathrm{A^{-1}=\frac{1}{7}\left (5I-A \right )}

Statement is 1 is true

\begin{aligned} \mathrm{A^{3}-2 A^{2}-3 A+I &=\mathrm{A(5 A-7 I)-2 A^{2}-3 A+I}} \\ &=\mathrm{5 A^{2}-7 A-2 A^{2}-3 A+I} \\ &=3 A^{2}-10A+I \\ &=3(5 A-7 I)-10 A+I \\ &=15 A-2 1I-10 A+I=5 A-20 I \\ &=5(A-4 I) \end{aligned}

Statement 2 also correct

Posted by

Devendra Khairwa

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