If `D_(1) and D_(2)` are two `3x3` diagnal matrices where none
of the diagonal elements is zero, then

Options

A. `D_(1) D_(2)` is a diagonal matrix
B. `D_(1) D_(2) = D_(2)D_(1)`
C. `D_(1)^(2)+D_(2)^(2)` is a diagonal matrix
D. None of the above

Explanation

To solve the problem, we need to analyze the properties of two 3x3 diagonal matrices \( D_1 \) and \( D_2 \), where none of the diagonal elements is zero.

### Step-by-Step Solution

1. **Define the Diagonal Matrices**:
Let \( D_1 \) and \( D_2 \) be defined as follows:
\[ D_1 = \begin{pmatrix} d_{11} & 0 & 0 \\ 0 & d_{12} & 0 \\ 0 & 0 & d_{13} \end{pmatrix}, \quad D_2 = \begin{pmatrix} d_{21} & 0 & 0 \\ 0 & d_{22} & 0 \\ 0 & 0 & d_{23} \end{pmatrix} \]
where \( d_{11}, d_{12}, d_{13}, d_{21}, d_{22}, d_{23} \) are all non-zero.

2. **Check if \( D_1 D_2 \) is a Diagonal Matrix**:
The product of two diagonal matrices is also a diagonal matrix. Therefore, we calculate:
\[ D_1 D_2 = \begin{pmatrix} d_{11} & 0 & 0 \\ 0 & d_{12} & 0 \\ 0 & 0 & d_{13} \end{pmatrix} \begin{pmatrix} d_{21} & 0 & 0 \\ 0 & d_{22} & 0 \\ 0 & 0 & d_{23} \end{pmatrix} = \begin{pmatrix} d_{11}d_{21} & 0 & 0 \\ 0 & d_{12}d_{22} & 0 \\ 0 & 0 & d_{13}d_{23} \end{pmatrix} \]
Since the product results in a diagonal matrix, this option is correct.

3. **Check if \( D_1 D_2 = D_2 D_1 \)**:
Since matrix multiplication is commutative for diagonal matrices, we have:
\[ D_2 D_1 = \begin{pmatrix} d_{21} & 0 & 0 \\ 0 & d_{22} & 0 \\ 0 & 0 & d_{23} \end{pmatrix} \begin{pmatrix} d_{11} & 0 & 0 \\ 0 & d_{12} & 0 \\ 0 & 0 & d_{13} \end{pmatrix} = \begin{pmatrix} d_{21}d_{11} & 0 & 0 \\ 0 & d_{22}d_{12} & 0 \\ 0 & 0 & d_{23}d_{13} \end{pmatrix} \]
Thus, \( D_1 D_2 = D_2 D_1 \), confirming this option is also correct.

4. **Check if \( D_1^2 + D_2^2 \) is a Diagonal Matrix**:
We calculate \( D_1^2 \) and \( D_2^2 \):
\[ D_1^2 = \begin{pmatrix} d_{11}^2 & 0 & 0 \\ 0 & d_{12}^2 & 0 \\ 0 & 0 & d_{13}^2 \end{pmatrix}, \quad D_2^2 = \begin{pmatrix} d_{21}^2 & 0 & 0 \\ 0 & d_{22}^2 & 0 \\ 0 & 0 & d_{23}^2 \end{pmatrix} \]
Now, adding these two matrices:
\[ D_1^2 + D_2^2 = \begin{pmatrix} d_{11}^2 + d_{21}^2 & 0 & 0 \\ 0 & d_{12}^2 + d_{22}^2 & 0 \\ 0 & 0 & d_{13}^2 + d_{23}^2 \end{pmatrix} \]
This is also a diagonal matrix since all off-diagonal elements are zero.

### Conclusion
All three options are correct:
- \( D_1 D_2 \) is a diagonal matrix.
- \( D_1 D_2 = D_2 D_1 \).
- \( D_1^2 + D_2^2 \) is a diagonal matrix.