To solve for [tex]\( 2D + E \)[/tex] given the matrices [tex]\( D \)[/tex] and [tex]\( E \)[/tex], we need to follow several steps:
1. Double matrix [tex]\( D \)[/tex]: Calculate [tex]\( 2D \)[/tex] by multiplying each element of matrix [tex]\( D \)[/tex] by 2.
2. Add matrix [tex]\( E \)[/tex] to [tex]\( 2D \)[/tex]: Perform matrix addition by adding the corresponding elements of matrices [tex]\( 2D \)[/tex] and [tex]\( E \)[/tex].
Let's break this down step-by-step:
### Step 1: Calculate [tex]\( 2D \)[/tex]
Given matrix [tex]\( D \)[/tex]:
[tex]\[ D = \begin{pmatrix} 1 & 2 \\ -1 & 3 \end{pmatrix} \][/tex]
We need to double each element of [tex]\( D \)[/tex]:
[tex]\[ 2D = 2 \times \begin{pmatrix} 1 & 2 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 2 \times 1 & 2 \times 2 \\ 2 \times -1 & 2 \times 3 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ -2 & 6 \end{pmatrix} \][/tex]
### Step 2: Add matrix [tex]\( E \)[/tex] to [tex]\( 2D \)[/tex]
Given matrix [tex]\( E \)[/tex]:
[tex]\[ E = \begin{pmatrix} 4 & 5 \\ 6 & -2 \end{pmatrix} \][/tex]
Now we add the corresponding elements of matrices [tex]\( 2D \)[/tex] and [tex]\( E \)[/tex]:
[tex]\[ 2D + E = \begin{pmatrix} 2 & 4 \\ -2 & 6 \end{pmatrix} + \begin{pmatrix} 4 & 5 \\ 6 & -2 \end{pmatrix} \][/tex]
Add the elements one by one:
- The element in the first row, first column: [tex]\( 2 + 4 = 6 \)[/tex]
- The element in the first row, second column: [tex]\( 4 + 5 = 9 \)[/tex]
- The element in the second row, first column: [tex]\( -2 + 6 = 4 \)[/tex]
- The element in the second row, second column: [tex]\( 6 + (-2) = 4 \)[/tex]
Therefore, the resulting matrix [tex]\( 2D + E \)[/tex] is:
[tex]\[ 2D + E = \begin{pmatrix} 6 & 9 \\ 4 & 4 \end{pmatrix} \][/tex]
### Conclusion
The doubled matrix [tex]\( 2D \)[/tex] and the resulting matrix [tex]\( 2D + E \)[/tex] are:
[tex]\[ 2D = \begin{pmatrix} 2 & 4 \\ -2 & 6 \end{pmatrix} \][/tex]
[tex]\[ 2D + E = \begin{pmatrix} 6 & 9 \\ 4 & 4 \end{pmatrix} \][/tex]
These steps show how we arrived at the final answer by doubling [tex]\( D \)[/tex] and then adding [tex]\( E \)[/tex] to get [tex]\( 2D + E \)[/tex].
