Suppose $X=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ satisfies the equation $X^2-4X+3I=O$

Statement-I: If $a+d\ne 4$ then there are just two such matrices X.

Statement –II: There are infinite numbers of matrices X, satisfying $X^2-4X+3I=O$

${\mathrm{X}}^2-4\mathrm{X}+3\mathrm{I}=\mathrm{O}\Rightarrow (\mathrm{X}-\mathrm{I})(\mathrm{X}-3\mathrm{I})=\mathrm{O}$

$\Rightarrow \left(\begin{array}{cc}(a-1)(a-3)+bc& b(a+d-4)\\ c(a+d-4)& (d-1)(d-3)+bc\end{array}\right)=\left(\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\end{array}\right)$ $$\begin{gathered} \text{ If }a+d\ne 4\text{, then }b=0,c=0 \\ (a-1)(a-3)=0,(\mathrm{d}-1)(\mathrm{d}-3)=0 \\ \Rightarrow \mathrm{a}=1,3,\mathrm{d}=1,3 \\ \text{ As }a+d\ne 4,a=1,d=1\text{ or }a=3,d=3\text{. } \end{gathered}$$

$$\begin{gathered} \text{ Ifa }+d=4\text{, we get } \\ \left(\begin{array}{cc}(a-1)(a-3)+bc& 0\\ 0& (a-1)(3-a)+bc\end{array}\right)=0 \end{gathered}$$

$\therefore$ There are infinite number of matrices satisfying  $X^2-4X+3I=O$