JEE Main2016MathematicsMatricesActual
Let A , be a 3 × 3 matrix, such that A 2 - 5 A + 7 I = O . Statement - I : A - 1 = 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 :
Options
- ABoth the statements are true
- BBoth the statements are false
- CStatement - I is true, but Statement - II is false
- DStatement - I is false, but Statement - II is true
Correct answer
A. Both the statements are true
Step-by-step solution
A 2 = 5 A - 7 I . . . . . i A A A - 1 - 5 A A - 1 - 7 I A - 1 = O A I = 5 I - 7 A - 1 A - 1 = 1 7 ( 5 I - A ) Hence, the statement I is true. Now A 3 - 2 A 2 - 3 A + I = A 5 A - 7 I - 2 A 2 - 3 A + I from i = 5 A 2 - 7 A - 2 A 2 - 3 A + I = 3 A 2 - 10 A + I = 3 5 A - 7 I - 10 A + I from i = 5 A - 20 I = 5 A - 4 I Hence, Statement II is true.