The number of values of `lambda` for which there exist a non -zero `3xx3` matrix A such that ` A = lambda A` is

Options

A. 0
B. 1
C. 2
D. infinite

Answer

To solve the problem, we need to find the number of values of \( \lambda \) for which there exists a non-zero \( 3 \times 3 \) matrix \( A \) such that \( A = \lambda A \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ A = \lambda A \]

2. **Rearrange the equation:**
\[ A - \lambda A = 0 \]
This simplifies to:
\[ (1 - \lambda) A = 0 \]

3. **Analyze the equation:**
Since \( A \) is a non-zero matrix, the only way for the product \( (1 - \lambda) A \) to equal the zero matrix is if the scalar factor \( (1 - \lambda) \) is zero. This leads us to:
\[ 1 - \lambda = 0 \]

4. **Solve for \( \lambda \):**
\[ \lambda = 1 \]

5. **Determine the number of values of \( \lambda \):**
The only value of \( \lambda \) that satisfies the equation is \( \lambda = 1 \). Therefore, there is exactly **one value** of \( \lambda \).

### Final Answer:
The number of values of \( \lambda \) for which there exists a non-zero \( 3 \times 3 \) matrix \( A \) such that \( A = \lambda A \) is **1**.

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