Mathematics - Matrices Question with Solution | TestHub

$A^2=5A-7I.....\left(i\right)$ 

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

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

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

Hence, the statement I is true.

Now $A^3-2A^2-3A+I$

$=A\left(5A-7I\right)-2A^2-3A+I$       {from$\left(i\right)$} 

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

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

$=3\left(5A-7I\right)-10A+I$       {from$\left(i\right)$}

$=5A-20I$

$=5\left(A-4I\right)$

Hence, Statement II is true.