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Let S={n in N ,[(0,i),(i,0)]^n[(a,b),(c,...

Let `S={n in N ,[(0,i),(i,0)]^n[(a,b),(c,d)]=[(a,b),(c,d)] forall a,b,c,d in R.` Find the number of 2-digit numbers in S

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To solve the problem, we need to analyze the given condition involving the matrix \( B = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \) raised to the power \( n \) and its relationship with the identity matrix. ### Step-by-Step Solution 1. **Understanding the Matrix Equation**: We start with the equation: \[ B^n \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] This implies that \( B^n \) must equal the identity matrix \( I \) when multiplied by any matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \). 2. **Finding \( B^n \)**: The matrix \( B \) can be expressed as: \[ B = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \] We can calculate the powers of \( B \): - \( B^1 = B \) - \( B^2 = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = -I \) - \( B^3 = B^2 \cdot B = -I \cdot B = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix} \) - \( B^4 = B^2 \cdot B^2 = (-I)(-I) = I \) From this, we see that \( B^4 = I \). 3. **Condition for \( B^n = I \)**: For \( B^n = I \), \( n \) must be a multiple of 4. Therefore, we can express this condition as: \[ n = 4k \quad \text{for some integer } k \] 4. **Finding Two-Digit Multiples of 4**: We need to find the two-digit natural numbers that are multiples of 4. The smallest two-digit number is 10, and the largest is 99. - The smallest two-digit multiple of 4 is 12 (since \( 4 \times 3 = 12 \)). - The largest two-digit multiple of 4 is 96 (since \( 4 \times 24 = 96 \)). 5. **Counting the Two-Digit Multiples of 4**: The sequence of two-digit multiples of 4 can be represented as: \[ 12, 16, 20, \ldots, 96 \] This is an arithmetic sequence where: - First term \( a = 12 \) - Common difference \( d = 4 \) To find the number of terms \( n \) in this sequence, we can use the formula for the \( n \)-th term of an arithmetic sequence: \[ a_n = a + (n-1)d \] Setting \( a_n = 96 \): \[ 96 = 12 + (n-1) \cdot 4 \] \[ 84 = (n-1) \cdot 4 \] \[ n-1 = 21 \quad \Rightarrow \quad n = 22 \] ### Final Answer The number of two-digit numbers in the set \( S \) is **22**.
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