Question

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

Answer 1

Answer: (C)

$A^{2}-5 A+7 I=0|A| \pm 0$
$\Rightarrow A -5 I =-7 A ^{-1}$
$\Rightarrow A ^{-1}=\frac{1}{7}(5 I - A )$
Hence statement $1$ is true
Now $A^{3}-2 A^{2}-3 A+I=A\left(A^{2}\right)-2 A^{2}-3 A+I$
$= A (5 A -7 I )-2 A ^{2}-3 A + I$
$=3 A ^{2}-10 A + I$
$=5 A -20 I $
$=3((5 A -7 I )-10 A + I$
$=5( A -4 I )$
Statement $2$ also correct