Let A be an n `xx` n matrix. If det `(lambda A) = lambda^(s)` det(A), what is the value of s?

Options

A. 0
B. 1
C. -1
D. n

Explanation

To solve the problem, we need to find the value of \( s \) in the equation given by:

\[ \text{det}(\lambda A) = \lambda^s \cdot \text{det}(A) \]

where \( A \) is an \( n \times n \) matrix.

### Step-by-step Solution:

1. **Understanding the Determinant of a Scalar Multiple of a Matrix**:
The determinant of a scalar multiple of a matrix can be expressed using the property:
\[ \text{det}(\lambda A) = \lambda^n \cdot \text{det}(A) \]
where \( n \) is the order (or size) of the square matrix \( A \).

2. **Setting Up the Equation**:
From the property mentioned above, we can rewrite the equation as:
\[ \text{det}(\lambda A) = \lambda^n \cdot \text{det}(A) \]

3. **Comparing Both Sides**:
We are given that:
\[ \text{det}(\lambda A) = \lambda^s \cdot \text{det}(A) \]
Now, we can set the two expressions for \(\text{det}(\lambda A)\) equal to each other:
\[ \lambda^n \cdot \text{det}(A) = \lambda^s \cdot \text{det}(A) \]

4. **Dividing Both Sides by \(\text{det}(A)\)**:
Assuming \(\text{det}(A) \neq 0\), we can divide both sides by \(\text{det}(A)\):
\[ \lambda^n = \lambda^s \]

5. **Equating the Exponents**:
Since this equality must hold for all values of \(\lambda\), we can equate the exponents:
\[ n = s \]

6. **Conclusion**:
Thus, the value of \( s \) is:
\[ s = n \]

### Final Answer:
The value of \( s \) is \( n \).

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