A square matrix `[a_(ij)]` such that `a_(ij)=0` for `i ne j and a_(ij) = k` where `k` is a constant for `i = j` is called _____

Options

A. (a) diagonal matrix but not scalar matrix
B. (b) scalar matrix
C. (c) unit matrix
D. (d) none of the above

Answer

To solve the question, we need to analyze the properties of the given square matrix defined by the conditions provided.

### Step-by-Step Solution:

1. **Understanding the Matrix Definition**:
The matrix \( A = [a_{ij}] \) is defined such that:
- \( a_{ij} = 0 \) for \( i \neq j \)
- \( a_{ij} = k \) (where \( k \) is a constant) for \( i = j \)

2. **Constructing the Matrix**:
Let's construct a 3x3 matrix as an example based on the given conditions:
\[ A = \begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix} \]
Here, all off-diagonal elements are zero, and the diagonal elements are equal to \( k \).

3. **Identifying the Type of Matrix**:
- A **diagonal matrix** is defined as a matrix where all off-diagonal elements are zero. Since all off-diagonal elements in our matrix \( A \) are zero, it qualifies as a diagonal matrix.
- A **scalar matrix** is a special type of diagonal matrix where all the diagonal elements are the same constant. In our case, all diagonal elements are equal to \( k \), which means \( A \) is also a scalar matrix.
- A **unit matrix** (or identity matrix) is a diagonal matrix where all diagonal elements are equal to 1. Since our diagonal elements are \( k \), which is not necessarily 1, \( A \) is not a unit matrix.

4. **Conclusion**:
Based on the definitions:
- The matrix \( A \) is a scalar matrix because all diagonal elements are equal to the constant \( k \) and all off-diagonal elements are zero.
- Therefore, the correct answer to the question is **option B: scalar matrix**.