Which one of the following is a scalar matrix

Options

A. `[(1,1),(1,1)]`
B. `[(6,0),(0,3)]`
C. `[(-8,0),(0,-8)]`
D. none of these

Explanation

To determine which of the given options is a scalar matrix, we first need to understand the definition of a scalar matrix.

### Definition of a Scalar Matrix:
A scalar matrix is a square matrix in which all the elements of the principal diagonal are the same (let's denote this common value as \( k \)), and all the non-diagonal elements are zero. Thus, a scalar matrix can be represented as:

\[ \begin{pmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{pmatrix} \]

where \( k \) is a non-zero scalar.

### Steps to Identify the Scalar Matrix:

1. **Examine Each Option:**
We will analyze each option provided to see if it meets the criteria of a scalar matrix.

2. **Check Diagonal Elements:**
For each option, we will check if all diagonal elements are equal.

3. **Check Non-Diagonal Elements:**
We will also check if all non-diagonal elements are zero.

### Step-by-Step Solution:

1. **Option A:**
- Suppose the matrix is:
\[ \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \]
- Diagonal elements: 1, 1 (equal)
- Non-diagonal elements: 1, 1 (not zero)
- **Conclusion:** Not a scalar matrix.

2. **Option B:**
- Suppose the matrix is:
\[ \begin{pmatrix} 6 & 0 \\ 3 & 6 \end{pmatrix} \]
- Diagonal elements: 6, 6 (equal)
- Non-diagonal elements: 0, 3 (not zero)
- **Conclusion:** Not a scalar matrix.

3. **Option C:**
- Suppose the matrix is:
\[ \begin{pmatrix} -8 & 0 \\ 0 & -8 \end{pmatrix} \]
- Diagonal elements: -8, -8 (equal)
- Non-diagonal elements: 0, 0 (zero)
- **Conclusion:** This is a scalar matrix.

4. **Option D:**
- Suppose the matrix is:
\[ \begin{pmatrix} 2 & 3 \\ 4 & 2 \end{pmatrix} \]
- Diagonal elements: 2, 2 (equal)
- Non-diagonal elements: 3, 4 (not zero)
- **Conclusion:** Not a scalar matrix.

### Final Answer:
The scalar matrix among the options is **Option C**.