Question:
Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
Updated On: Mar 17, 2025
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Correct Answer: 16
Solution and Explanation
Expression for the Determinant of A
The determinant of matrix \( A \) is given as:
\[ A = \begin{bmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{bmatrix} \]
The determinant is computed as:
\[ |A| = (d - e) + c(b - a) \]
Conditions for \( |A| \in \{-1, 1\} \)
The determinant must satisfy:
\[ |A| = \pm 1 \]
Analysis of Possible Values of \( a, b, c, d, e \)
- Each variable \( a, b, c, d, e \) takes values in \(\{0, 1\}\).
- The total number of possible matrices is \( 2^5 = 32 \).
- We select only those matrices where \( |A| = \pm 1 \).
Case 1: \( |A| = 1 \)
From equation:
\[ (d - e) + c(b - a) = 1 \]
Valid combinations:
- For \( c = 0 \), \( d - e = 1 \), \( b - a = 0 \).
- For \( c = 1 \), \( d - e = 0 \), \( b - a = 1 \).
Case 2: \( |A| = -1 \)
From equation:
\[ (d - e) + c(b - a) = -1 \]
Valid combinations:
- For \( c = 0 \), \( d - e = -1 \), \( b - a = 0 \).
- For \( c = 1 \), \( d - e = 0 \), \( b - a = -1 \).
Total Valid Matrices
For each case (\( |A| = 1 \) and \( |A| = -1 \)), there are 8 valid matrices. Thus, the total number of matrices in \( S \) is:
\[ 8 + 8 = 16 \]
Final Answer: 16
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