Let S = left{ begin{bmatrix} a_{11} & a_{12} | vedclass

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(D) matrix $A = \begin{bmatrix} a & b \ c & a \end{bmatrix}$ is in $S$ where $a, b, c \in \{0, 1, 2\}$.There are $3$ choices for each of $a, b, c$,so

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$20$

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(D) matrix $A = \begin{bmatrix} a & b \ c & a \end{bmatrix}$ is in $S$ where $a, b, c \in \{0, 1, 2\}$.There are $3$ choices for each of $a, b, c$,so the total number of matrices in $S$ is $3 	imes 3 	imes 3 = 27$.For a matrix to be singular,its determinant must be zero: $\det(A) = a^2 - bc = 0$,which implies $a^2 = bc$.We check the cases for $a \in \{0, 1, 2\}$:Case $1$: $a = 0$. Then $bc = 0$. The pairs $(b, c)$ can be $(0, 0), (0, 1), (0, 2), (1, 0), (2, 0)$. There are $5$ such matrices.Case $2$: $a = 1$. Then $bc = 1$. The only pair $(b, c)$ is $(1, 1)$. There is $1$ such matrix.Case $3$: $a = 2$. Then $bc = 4$. The only pair $(b, c)$ is $(2, 2)$. There is $1$ such matrix.Total singular matrices $= 5 + 1 + 1 = 7$.Number of non-singular matrices $= 	ext{Total} - 	ext{Singular} = 27 - 7 = 20$.

Let $S = \left\{ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} : a_{ij} \in \{0, 1, 2\}, a_{11} = a_{22} \right\}$. Then the number of non-singular matrices in the set $S$ is

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