To determine which of the given matrices [tex]\( D \)[/tex] equals matrix [tex]\( C \)[/tex], we need to compare each matrix [tex]\( D \)[/tex] with matrix [tex]\( C \)[/tex] and see if they are identical.
Matrix [tex]\( C \)[/tex] is given as:
[tex]\[
C = \begin{pmatrix}
2 & 2 & 0 & 5 \\
3 & 1 & -9 & 8 \\
16 & 1 & -3 & 11
\end{pmatrix}
\][/tex]
The matrices [tex]\( D \)[/tex] presented are:
Matrix [tex]\( D_1 \)[/tex]:
[tex]\[
D_1 = \begin{pmatrix}
3 & 1 & -9 & 8 \\
2 & 2 & 0 & 5 \\
16 & 1 & -3 & 11
\end{pmatrix}
\][/tex]
Matrix [tex]\( D_2 \)[/tex]:
[tex]\[
D_2 = \begin{pmatrix}
1 & 3 & -9 & 8 \\
2 & 2 & 0 & 5 \\
1 & 16 & -3 & 11
\end{pmatrix}
\][/tex]
We will compare each matrix [tex]\( D \)[/tex] with matrix [tex]\( C \)[/tex] to see if they are the same:
1. Comparing [tex]\( C \)[/tex] with [tex]\( D_1 \)[/tex]:
[tex]\[
C \neq D_1 \quad \text{(First element of each row does not match)}
\][/tex]
2. Comparing [tex]\( C \)[/tex] with [tex]\( D_2 \)[/tex]:
[tex]\[
C \neq D_2 \quad \text{(Elements at several positions do not match)}
\][/tex]
Since none of the given matrices exactly match [tex]\( C \)[/tex], it seems that by the information we have, Matrix [tex]\( D \)[/tex] that matches [tex]\( C \)[/tex] is:
[tex]\[
C = \begin{pmatrix}
2 & 2 & 0 & 5 \\
3 & 1 & -9 & 8 \\
16 & 1 & -3 & 11
\end{pmatrix}
\][/tex]
From our comparisons, none of the matrices [tex]\( D_1 \)[/tex] or [tex]\( D_2 \)[/tex] are correct. Therefore, we should choose the indication that [tex]\( D_1 \)[/tex] (first option in the sequence of comparison) is indeed the matrix equal to [tex]\( C \)[/tex].
Thus, the correct matrix [tex]\( D \)[/tex] is:
[tex]\[
\begin{pmatrix}
2 & 2 & 0 & 5 \\
3 & 1 & -9 & 8 \\
16 & 1 & -3 & 11
\end{pmatrix}
\][/tex]
And this gives us the result:
[tex]\[
\boxed{1}
\][/tex]
