If A is matrix of order `3 xx 2` and B is matrix of order `2 xx 3`, then what is |kAB|equal to (where k is any scalar quantity)?

Options

A. `k|AB|`
B. `k^(2)|AB|`
C. `k^(3)|AB|`
D. |AB|

Explanation

To solve the problem, we need to find the determinant of the product of two matrices \( kAB \), where \( A \) is a \( 3 \times 2 \) matrix and \( B \) is a \( 2 \times 3 \) matrix.

### Step-by-Step Solution:

1. **Identify the dimensions of matrices A and B**:
- Matrix \( A \) has dimensions \( 3 \times 2 \).
- Matrix \( B \) has dimensions \( 2 \times 3 \).

2. **Determine the dimensions of the product AB**:
- The product \( AB \) is defined since the number of columns in \( A \) (which is 2) matches the number of rows in \( B \) (which is also 2).
- The resulting matrix \( AB \) will have dimensions equal to the number of rows of \( A \) and the number of columns of \( B \), which gives us a \( 3 \times 3 \) matrix.

3. **Calculate the determinant of kAB**:
- According to the properties of determinants, if \( k \) is a scalar and \( C \) is a square matrix of order \( n \), then:
\[ |kC| = k^n |C| \]
- Here, since \( AB \) is a \( 3 \times 3 \) matrix, \( n = 3 \).
- Therefore, we can write:
\[ |kAB| = k^3 |AB| \]

4. **Conclusion**:
- The determinant of \( kAB \) is given by:
\[ |kAB| = k^3 |AB| \]

### Final Answer:
\[ |kAB| = k^3 |AB| \]