Let A be a 3 × 3 matrix such that A2-5A+7I=O . Statement - I: A-1 = (1/7) (5 I - A). Statement - II: The polynomial A3 - 2A2 - 3A + I can be reduced to 5 (A - 4 I). Then :

Question

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

Tardigrade Admin · Accepted Answer

A2-5 A+7 I=0|A| ± 0 ⇒ A -5 I =-7 A -1 ⇒ A -1=(1/7)(5 I - A ) Hence statement 1 is true Now A3-2 A2-3 A+I=A(A2)-2 A2-3 A+I = A (5 A -7 I )-2 A 2-3 A + I =3 A 2-10 A + I =5 A -20 I =3((5 A -7 I )-10 A + I =5( A -4 I ) Statement 2 also correct

Q. Let A be a 3×3 matrix such that A2−5A+7I=O . Statement - I : A−1=71​(5I−A). Statement - II : The polynomial A3−2A2−3A+I can be reduced to 5(A−4I). Then :

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.

A2−5A+7I=0∣A∣±0 ⇒A−5I=−7A−1 ⇒A−1=71​(5I−A) Hence statement 1 is true Now A3−2A2−3A+I=A(A2)−2A2−3A+I =A(5A−7I)−2A2−3A+I =3A2−10A+I =5A−20I =3((5A−7I)−10A+I =5(A−4I) Statement 2 also correct

Q. Let $A$ be a $3 \times 3$ matrix such that $A^{2} - 5 A + 7 I = O$ .
Statement - I : $A^{- 1} = \frac{1}{7} (5 I - A)$ .
Statement - II : The polynomial $A^{3} - 2 A^{2} - 3 A + I$ can be reduced to $5 (A - 4 I)$ . Then :