Let `S={((-1,a),(0,b)):a ,b in {1,2,3......100} and " let" T_(n) - {A in S : A^(n(n+1))=I}` .Then the number of elements in `cap_(n=1)^(100) T_(n)` is _______.

Answer

To solve the problem, we need to analyze the set \( S \) and the conditions for the matrices in \( T_n \).

### Step 1: Define the Matrix
The matrices in the set \( S \) are of the form:
\[ A = \begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix} \]
where \( a, b \) are integers from 1 to 100.

### Step 2: Determine the Condition for \( T_n \)
The set \( T_n \) is defined as:
\[ T_n = \{ A \in S : A^{n(n+1)} = I \} \]
where \( I \) is the identity matrix. We need to find the values of \( n \) such that \( A^{n(n+1)} = I \).

### Step 3: Calculate \( A^2 \)
First, we compute \( A^2 \):
\[ A^2 = A \cdot A = \begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix} \cdot \begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix} = \begin{pmatrix} 1 & -a + ab \\ 0 & b^2 \end{pmatrix} \]
For \( A^2 = I \), we need:
\[ \begin{pmatrix} 1 & -a + ab \\ 0 & b^2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
This gives us the equations:
1. \( -a + ab = 0 \)
2. \( b^2 = 1 \)

### Step 4: Solve the Equations
From \( b^2 = 1 \), we find \( b = 1 \) (since \( b \) must be in \( \{1, 2, \ldots, 100\} \)).

Substituting \( b = 1 \) into the first equation:
\[ -a + a \cdot 1 = 0 \implies -a + a = 0 \implies 0 = 0 \]
This is always true for any \( a \).

### Step 5: Calculate Higher Powers
Next, we calculate \( A^4 \):
\[ A^4 = A^2 \cdot A^2 = \begin{pmatrix} 1 & -a + ab \\ 0 & b^2 \end{pmatrix} \cdot \begin{pmatrix} 1 & -a + ab \\ 0 & b^2 \end{pmatrix} \]
Substituting \( b = 1 \):
\[ A^4 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
This means \( A^4 = I \).

### Step 6: Generalizing the Powers
We find that:
- \( A^2 = I \) when \( n(n+1) = 2 \)
- \( A^4 = I \) when \( n(n+1) = 4 \)
- \( A^{12} = I \) when \( n(n+1) = 12 \)

### Step 7: Finding \( n \)
We need to find \( n \) such that:
- \( n(n+1) = 2 \) gives \( n = 1 \)
- \( n(n+1) = 4 \) gives \( n = 2 \)
- \( n(n+1) = 12 \) gives \( n = 3 \)

### Step 8: Intersection of \( T_n \)
We need to find the intersection \( \bigcap_{n=1}^{100} T_n \). The only valid \( A \) that satisfies all conditions is:
\[ A = \begin{pmatrix} -1 & a \\ 0 & 1 \end{pmatrix} \]
where \( a \) can take any value from 1 to 100.

### Final Count
Thus, the number of elements in \( \bigcap_{n=1}^{100} T_n \) is \( 100 \) (corresponding to the values of \( a \)).

### Answer
The number of elements in \( \bigcap_{n=1}^{100} T_n \) is **100**.