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
$A^3-2A^2-3A+Icanbereduceto5(A-4I)$

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

Answer 1

The correct option is C Both the statement are true

$A^2-5A+7I$ $\Rightarrow$ $A^2=5A-7I$

$\Rightarrow$ $AA{A}^{-1}=5A{A}^{-1}-7I{A}^{-1}$

$\Rightarrow$ $AI=5I-7A^{-1}$

$\Rightarrow$ $A^{-1}=\frac{1}{7}(5I-A)$

$\therefore$ Statement $I$ is true.

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

$=5A^2-7A-2A^2-3A+I$

$=3A^2-10A+I$

$=3(5A-7I)-10A+I$

$=5A-20I$

$=5(A-4I)$

$\therefore$ Statement $II$ is also true.