If two sets A and B are having 99 elements in common, the number of elements common to each of the sets `A xx B` and `B xx A` are `121 lambda^(2)`, the value of `lambda` is

Explanation

To solve the problem step by step, we will analyze the information given and apply the relevant mathematical concepts.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We are given two sets A and B that have 99 elements in common. We need to find the value of λ, given that the number of elements common to the sets \( A \times B \) and \( B \times A \) is \( 121\lambda^2 \).

2. **Using the Theorem:**
We can use the theorem which states:
\[ A \times B \cap C \times D = A \cap C \times B \cap D \]
In our case, we can set \( C = B \) and \( D = A \). Thus, we can rewrite the intersection as:
\[ A \times B \cap B \times A = A \cap B \times A \cap B \]

3. **Finding the Cardinality:**
The cardinality (number of elements) of the intersection \( A \times B \cap B \times A \) can be expressed as:
\[ |A \times B \cap B \times A| = |A \cap B| \times |A \cap B| \]
This simplifies to:
\[ |A \times B \cap B \times A| = (|A \cap B|)^2 \]

4. **Substituting Known Values:**
We know that \( |A \cap B| = 99 \) (the number of common elements in sets A and B). Therefore:
\[ |A \times B \cap B \times A| = 99^2 \]

5. **Setting Up the Equation:**
We are given that:
\[ |A \times B \cap B \times A| = 121\lambda^2 \]
Thus, we can set up the equation:
\[ 99^2 = 121\lambda^2 \]

6. **Solving for λ:**
To find λ, we rearrange the equation:
\[ \lambda^2 = \frac{99^2}{121} \]
Since \( 121 = 11^2 \), we can rewrite this as:
\[ \lambda^2 = \frac{99^2}{11^2} = \left(\frac{99}{11}\right)^2 \]
Therefore:
\[ \lambda = \frac{99}{11} = 9 \quad \text{or} \quad \lambda = -9 \]

7. **Final Answer:**
The possible values of λ are:
\[ \lambda = 9 \quad \text{or} \quad \lambda = -9 \]