Question

Let A be a $3\times 3$ 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:

• A. Both the statements are true
• B. Both the statements are false
• C. Statement - I is true, but Statement - II is false
• D. Statement - I is false, but Statement - II is true

Answer 1

The correct option is A Both the statements are true

Given

$A^2=5A-7I$
$AA{A}^{-1}=5A{A}^{-1}-7I{A}^{-1}$
$AI=5I-7A^{-1}$
$AI=5I-7A^{-1}$
$A^{-1}=\frac{1}{7}(5I-A)$

Statement I is true.

$A^3-2A^2-3A+I=A(5A-7I)-2A^2-3A+I$
$=5A^2-7A-2A^2-3A+I$
$=3A^2-10A+I$
$=3(5A-7I)-10A+I$
$=5A-20I$
$=5(A-4I)$

Statement II is also true.