Select the correct answer from the drop-down menu.

[tex]\[
A=\left[\begin{array}{cc}
\sqrt{2} & 1 \\
1 & \sqrt{2} \\
\sqrt{2} & 1
\end{array}\right],
B=\left[\begin{array}{lll}
2 & 1 & 2 \\
1 & 2 & 1
\end{array}\right],
C=\left[\begin{array}{ll}
2 & 1 \\
1 & 2 \\
2 & 1
\end{array}\right],
D=\left[\begin{array}{cc}
\sqrt{4} & 1 \\
1 & \sqrt{4} \\
\sqrt{4} & 1
\end{array}\right]
\][/tex]

[tex]\(\square\)[/tex] are equal matrices.



Answer :

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.