To multiply matrix [tex]\( D \)[/tex] by matrix [tex]\( E \)[/tex], we need to follow the rules of matrix multiplication. Matrix [tex]\( D \)[/tex] is a [tex]\( 3 \times 1 \)[/tex] matrix and matrix [tex]\( E \)[/tex] is a [tex]\( 1 \times 2 \)[/tex] matrix. When multiplying an [tex]\( m \times n \)[/tex] matrix by an [tex]\( n \times p \)[/tex] matrix, the result is an [tex]\( m \times p \)[/tex] matrix.
Given:
[tex]\[
D=\left[\begin{array}{r}
5 \\
-2 \\
1
\end{array}\right]
\][/tex]
[tex]\[
E=\left[\begin{array}{ll}
1 & 2
\end{array}\right]
\][/tex]
Our goal is to compute the product [tex]\( D \times E \)[/tex].
1. Take the first element of [tex]\( D \)[/tex], which is [tex]\( 5 \)[/tex], and multiply by the first row of [tex]\( E \)[/tex]:
[tex]\[ 5 \times 1 = 5 \][/tex]
[tex]\[ 5 \times 2 = 10 \][/tex]
This results in the first row of the product matrix being [tex]\([5, 10]\)[/tex].
2. Take the second element of [tex]\( D \)[/tex], which is [tex]\( -2 \)[/tex], and multiply by the first row of [tex]\( E \)[/tex]:
[tex]\[ -2 \times 1 = -2 \][/tex]
[tex]\[ -2 \times 2 = -4 \][/tex]
This results in the second row of the product matrix being [tex]\([-2, -4]\)[/tex].
3. Take the third element of [tex]\( D \)[/tex], which is [tex]\( 1 \)[/tex], and multiply by the first row of [tex]\( E \)[/tex]:
[tex]\[ 1 \times 1 = 1 \][/tex]
[tex]\[ 1 \times 2 = 2 \][/tex]
This results in the third row of the product matrix being [tex]\([1, 2]\)[/tex].
Combining these rows, the product matrix is:
[tex]\[
\left[\begin{array}{rr}
5 & 10 \\
-2 & -4 \\
1 & 2
\end{array}\right]
\][/tex]
Therefore, the correct answer is:
[tex]\[
\text{D.} \left[\begin{array}{rr}
5 & 10 \\
-2 & -4 \\
1 & 2
\end{array}\right]
\][/tex]
