If A is a square matrix of order `3 and |3A|=k|A|` then find the value of k,

Answer

To solve the problem, we need to find the value of \( k \) given that \( |3A| = k|A| \) for a square matrix \( A \) of order 3.

### Step-by-Step Solution:

1. **Understand the Determinant Property**:
For any square matrix \( A \) of order \( n \), the determinant of a scalar multiple of the matrix can be expressed as:
\[ |kA| = k^n |A| \]
where \( k \) is a scalar and \( n \) is the order of the matrix.

2. **Identify the Order of the Matrix**:
In this case, the order of matrix \( A \) is 3 (i.e., \( n = 3 \)).

3. **Apply the Determinant Property**:
We substitute \( k = 3 \) and \( n = 3 \) into the determinant property:
\[ |3A| = 3^3 |A| \]

4. **Calculate \( 3^3 \)**:
\[ 3^3 = 27 \]
Therefore, we have:
\[ |3A| = 27 |A| \]

5. **Set Up the Equation**:
From the problem statement, we know:
\[ |3A| = k |A| \]
Equating the two expressions for \( |3A| \):
\[ 27 |A| = k |A| \]

6. **Solve for \( k \)**:
Assuming \( |A| \neq 0 \) (since if \( |A| = 0 \), the equation holds for any \( k \)), we can divide both sides by \( |A| \):
\[ k = 27 \]

### Final Answer:
The value of \( k \) is \( 27 \).

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