Let A , be a 3   × 3 matrix, such that A 2 - 5 A + 7 I = O . Statement - I : A - 1 = 1 7   5…

Let A, be a 3 ×3 matrix, such that A2-5A+7I=O.
Statement - I : A-1=17 5I-A.
Statement - II : The polynomial A3-2A2-3A+I,can be reduced to 5A-4I. Then :
  1. Both the statements are true
  2. Both the statements are false
  3. Statement - I is true, but Statement - II is false
  4. Statement - I is false, but Statement - II is true

Solution

A2=5A-7I.....i 

AAA-1-5AA-1-7IA-1=O

AI=5I-7A-1

A-1=17(5I-A)

Hence, the statement I is true.

Now A3-2A2-3A+I

=A5A-7I-2A2-3A+I       {fromi} 

=5A2-7A-2A2-3A+I

=3A2-10A+I

=35A-7I-10A+I       {fromi}

=5A-20I

=5A-4I

Hence, Statement II is true.

Asked in: JEE Main 2016 (10 Apr Online)

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