To determine which matrices are equal, let's examine matrices [tex]\( A \)[/tex] and [tex]\( D \)[/tex] in detail.
Matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
\sqrt{2} & 1 \\
1 & \sqrt{2} \\
\sqrt{2} & 1
\end{pmatrix}
\][/tex]
Matrix [tex]\( D \)[/tex]:
[tex]\[
D = \begin{pmatrix}
\sqrt{4} & 1 \\
1 & \sqrt{4} \\
\sqrt{4} & 1
\end{pmatrix}
\][/tex]
To compare matrices [tex]\( A \)[/tex] and [tex]\( D \)[/tex], first notice the values of the elements in each matrix:
For matrix [tex]\( A \)[/tex]:
- The first element is [tex]\(\sqrt{2}\)[/tex].
- The second element is 1.
- The third element is [tex]\( \sqrt{2} \)[/tex].
For matrix [tex]\( D \)[/tex]:
- The first element is [tex]\(\sqrt{4}\)[/tex], which simplifies to 2.
- The second element is 1.
- The third element is [tex]\(\sqrt{4}\)[/tex], which simplifies to 2.
Clearly, comparing the elements of matrices [tex]\( A \)[/tex] and [tex]\( D \)[/tex]:
- In the first matrix, the elements include [tex]\(\sqrt{2}\)[/tex].
- In the second matrix, the corresponding elements are 2.
Since the value of [tex]\(\sqrt{2}\)[/tex] (approximately 1.414) is not equal to 2, we conclude that matrices [tex]\( A \)[/tex] and [tex]\( D \)[/tex] are not identical element-wise. Therefore:
[tex]\[
A \neq D
\][/tex]
Thus, the correct answer to which matrices are equal is:
None are equal. The drop-down should reflect that no matrices are equal, confirming that [tex]\( A \)[/tex] and [tex]\( D \)[/tex] are not equal matrices.
