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Matrix Addition is a fundamental operation where two matrices of the same dimensions are combined by adding their corresponding elements. For example, if we have two 2x2 matrices:

• Matrix A: \[\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\]
• Matrix B: \[\begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]

Then, the sum of these matrices, \(A + B\), is computed as:\[\begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\]
In the given problem, after negating Matrix C and multiplying Matrix D by 2, the matrices were added:

• Negated C: \[\begin{bmatrix} 1 & -4 & 7 \ -2 & 6 & -11 \end{bmatrix}\]
• Matrix 2D: \[\begin{bmatrix} 14 & 18 & -12 \ -8 & 0 & 16 \end{bmatrix}\]

Each corresponding element was added to form the resulting matrix. This operation underlines the importance of ensuring matrices have the same dimensions, which allows each element to align perfectly and be summed correctly.

Scalar multiplication in matrices involves multiplying every element of a matrix by a constant, known as the scalar. This operation maintains the original layout of the matrix, only changing the values inside based on the scalar applied.
Let's consider a matrix \(D\):\[\begin{bmatrix} d_{11} & d_{12} & d_{13} \ d_{21} & d_{22} & d_{23} \end{bmatrix}\]If we need to find \(2D\), we multiply each element by 2:\[\begin{bmatrix} 2 \cdot d_{11} & 2 \cdot d_{12} & 2 \cdot d_{13} \ 2 \cdot d_{21} & 2 \cdot d_{22} & 2 \cdot d_{23} \end{bmatrix}\]
In this exercise, multiplying every element of Matrix D by the scalar 2 transformed it into:

• \[\begin{bmatrix} 14 & 18 & -12 \ -8 & 0 & 16 \end{bmatrix}\]

Scalar multiplication is useful when you need to scale a matrix up or down while retaining the positional integrity of the elements.

Matrix Negation is akin to multiplying all elements of a matrix by -1. This operation flips the sign of each element within the matrix, providing a complementary perspective for addition or subtraction tasks.
To negate a matrix like \(C\):\[\begin{bmatrix} c_{11} & c_{12} & c_{13} \ c_{21} & c_{22} & c_{23} \end{bmatrix}\]The negated matrix, \(-C\), is represented as:\[\begin{bmatrix} -c_{11} & -c_{12} & -c_{13} \ -c_{21} & -c_{22} & -c_{23} \end{bmatrix}\]
In practice, negating Matrix C turned it into:

• \[\begin{bmatrix} 1 & -4 & 7 \ -2 & 6 & -11 \end{bmatrix}\]

This process is foundational in combining matrices with addition, as it allows for the easy cancellation of values or the creation of "opposite" matrices for further computations.