If `|A_(n xx n)|=3 and |"adj A"|=243`, what is the value of n?

Options

A. 4
B. 5
C. 6
D. 7

Explanation

To solve the problem, we need to use the properties of determinants, specifically the relationship between the determinant of a matrix and the determinant of its adjoint.

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given:
- \( |A| = 3 \)
- \( |\text{adj } A| = 243 \)

2. **Use the Property of Determinants**:
The determinant of the adjoint of a matrix \( A \) is given by the formula:
\[ |\text{adj } A| = |A|^{n-1} \]
where \( n \) is the order of the matrix \( A \).

3. **Substitute the Known Values**:
We can substitute the known values into the formula:
\[ 243 = 3^{n-1} \]

4. **Express 243 as a Power of 3**:
We know that:
\[ 243 = 3^5 \]
Therefore, we can rewrite the equation as:
\[ 3^{n-1} = 3^5 \]

5. **Set the Exponents Equal**:
Since the bases are the same, we can set the exponents equal to each other:
\[ n - 1 = 5 \]

6. **Solve for \( n \)**:
Adding 1 to both sides gives:
\[ n = 6 \]

### Final Answer:
The value of \( n \) is \( 6 \).

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