Let S = left{ A = begin{bmatrix} 0 & 1 & c 1 | vedclass

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(C) The determinant of matrix $A$ is given by:$|A| = 0(ae - bd) - 1(e - d) + c(b - a) = c(b - a) + d - e$.We are given $a, b, c, d, e \in \{0, 1\}$ an

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$16$

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(C) The determinant of matrix $A$ is given by:$|A| = 0(ae - bd) - 1(e - d) + c(b - a) = c(b - a) + d - e$.We are given $a, b, c, d, e \in \{0, 1\}$ and $|A| \in \{-1, 1\}$.Case $I$: $c = 0$.Then $|A| = d - e$. For $|A| \in \{-1, 1\}$,we must have $(d, e) = (0, 1)$ or $(d, e) = (1, 0)$.For each pair $(d, e)$,there are $2 	imes 2 = 4$ choices for $(a, b)$.Total cases for $c = 0$ is $2 	imes 4 = 8$.Case $II$: $c = 1$.Then $|A| = (b - a) + (d - e)$. We need $(b - a) + (d - e) \in \{-1, 1\}$.Let $X = b - a$ and $Y = d - e$. $X, Y \in \{-1, 0, 1\}$.We need $X + Y \in \{-1, 1\}$.Possible values for $(X, Y)$ are:$(0, 1), (0, -1), (1, 0), (-1, 0), (1, 1) 	ext{ (sum 2, reject)}, (-1, -1) 	ext{ (sum -2, reject)}, (1, -1) 	ext{ (sum 0, reject)}, (-1, 1) 	ext{ (sum 0, reject)}$.Valid $(X, Y)$ pairs are $(0, 1), (0, -1), (1, 0), (-1, 0)$.For $X = 0$,$(a, b) \in \{(0, 0), (1, 1)\}$ ($2$ choices).For $X = 1$,$(a, b) = (0, 1)$ ($1$ choice).For $X = -1$,$(a, b) = (1, 0)$ ($1$ choice).Similarly for $Y$,$Y = 1 \implies (d, e) = (1, 0)$ ($1$ choice),$Y = -1 \implies (d, e) = (0, 1)$ ($1$ choice),$Y = 0 \implies (d, e) \in \{(0, 0), (1, 1)\}$ ($2$ choices).Total cases for $c = 1$: $(2 	imes 1) + (2 	imes 1) + (1 	imes 2) + (1 	imes 2) = 2 + 2 + 2 + 2 = 8$.Total elements in $S = 8 + 8 = 16$.

Let $S = \left\{ A = \begin{bmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{bmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\} \right\}$,where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is:

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