Let A be set of all $2 imes 2$ matrices of the form $\left(\begin{array}{ll}a & b \ c & a\end{arr

Question

Let A be set of all $2 	imes 2$ matrices of the form $\left(\begin{array}{ll}a & b \ c & a\end{array}ight)$, such that $a, b, c \in\{0,1,2,3,4\}$ then:

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For symmetric $x=\left(\begin{array}{ll}a & b \ b & a\end{array}ight) \Rightarrow 5^2$ matrices For skew symmetric $\left(\begin{array}{cc}0 & b \ -b & 0\end{array}ight)$ i.e. $\left(\begin{array}{ll}0 & 0 \ 0 & 0\end{array}ight)$ only possibility, already included in symmetric matrices. $$ |X|=a^2-b^2=(a+b)(a-b), a, b \in\{0,1,2,3,4\} $$ For it to be divisible by 5 either $a-b=0$ i.e. $a=b$ $$ \left(\begin{array}{ll} a & a \ a & a \end{array}ight) 	ext { i.e. } 5 	ext { matrices or } a+b=5 	ext { i.e. }\left(\begin{array}{ll} 1 & 4 \ 4 & 1 \end{array}ight) 	ext { or }\left(\begin{array}{ll} 2 & 3 \ 3 & 2 \end{array}ight) 	ext { or }\left(\begin{array}{ll} 3 & 2 \ 2 & 3 \end{array}ight) 	ext { or }\left(\begin{array}{ll} 4 & 1 \ 1 & 4 \end{array}ight) $$ Trace of $\left(\begin{array}{ll}a & b \ c & a\end{array}ight)$ is $2 a$, not divisible by $5 \Rightarrow a 
eq 0$ Determinant is $a^2-b c$ for it to be divisible by $5 b 
eq 0, c 
eq 0 \Rightarrow a, b, c \in\{1,2,3,4\}$ For any set of values of $a, b \in\{1,2,3,4\}$ we get a unique value of $c \Rightarrow 16$ such matrices. Determinant is $a^2-b c$ for it to be divisible by $5 b 
eq 0, c 
eq 0 \Rightarrow a, b, c \in\{1,2,3,4\}$ For any set of values of $a, b \in\{1,2,3,4\}$ we get a unique value of $c \Rightarrow 16$ such matrices. $$ \begin{aligned} & \left(\begin{array}{ll} 1 & 1 \ 1 & 1 \end{array}ight) \cdot\left(\begin{array}{ll} 2 & 2 \ 2 & 2 \end{array}ight) \cdot\left(\begin{array}{ll} 3 & 3 \ 3 & 3 \end{array}ight) \cdot\left(\begin{array}{ll} 4 & 4 \ 4 & 4 \end{array}ight) \cdot\left(\begin{array}{ll} 1 & 4 \ 4 & 1 \end{array}ight) \cdot\left(\begin{array}{ll} 2 & 3 \ 3 & 2 \end{array}ight) \cdot\left(\begin{array}{ll} 3 & 2 \ 2 & 3 \end{array}ight) \cdot\left(\begin{array}{ll} 4 & 1 \ 1 & 4 \end{array}ight) \ & \left(\begin{array}{ll} 1 & 2 \ 3 & 1 \end{array}ight) \cdot\left(\begin{array}{ll} 1 & 3 \ 2 & 1 \end{array}ight),\left(\begin{array}{ll} 2 & 4 \ 1 & 2 \end{array}ight) \cdot\left(\begin{array}{ll} 4 & 2 \ 3 & 4 \end{array}ight) \cdot\left(\begin{array}{ll} 2 & 1 \ 4 & 2 \end{array}ight) \cdot\left(\begin{array}{ll} 4 & 3 \ 2 & 4 \end{array}ight) \cdot\left(\begin{array}{ll} 3 & 1 \ 4 & 3 \end{array}ight) \cdot\left(\begin{array}{ll} 3 & 4 \ 1 & 3 \end{array}ight) \end{aligned} \ $$ Number of matrices in A whose determinant is not divisible by 5 = total - those which have determinant divisible by $5=5^3-(16+9)=100$ \ 9 comprises of $a=0, b=0, c 
eq 0$ i.e. $4 ; a=0, b 
eq 0, c=0$ i.e. $4 ; a=0, b=0, c=0$ i.e. 1