91Ó°ÊÓ

Matrix addition is a fundamental operation in linear algebra. It involves adding together two matrices of the same dimensions by adding their corresponding elements. For example, if we have matrix \(A\) and matrix \(B\), both of size \(n \times n\):

\[ A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ \text{...} & \text{...} \ \text{...} & \text{...} \end{pmatrix}, \, B = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \ \text{...} & \text{...} \ \text{...} & \text{...} \end{pmatrix} \]

Matrix addition would be performed as follows:

\[ A + B = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \ \text{...} & \text{...} \ \text{...} & \text{...} \end{pmatrix} \]

If matrix \(A\) and matrix \(B\) both have elements of the same dimensions, the operation will result into a new matrix of the same dimensions.

Matrix negation refers to the process of taking the negative of each element within a matrix. If matrix \(A\) has elements \(a_{ij}\), the negation of this matrix is written as \((-A)\) and will have elements \(-a_{ij}\). This is done element-wise. For instance:

\[ A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ \text{...} & \text{...} \ \text{...} & \text{...} \end{pmatrix} \implies -A = \begin{pmatrix} -a_{11} & -a_{12} \ -a_{21} & -a_{22} \ \text{...} & \text{...} \ \text{...} & \text{...} \end{pmatrix} \]

When you add matrix \(A\) and its negation \(-A\), the elements of the resulting matrix are all zero. This is because any number added to its negative will equal zero:

\[ a_{ij} + (-a_{ij}) = 0 \]
Which leads us to our final concept.

A Zero matrix, often denoted by \(0\), is a matrix in which all its elements are zero. The size of a zero matrix should match the size of the original matrices it's associated with. For example, a zero matrix for a 2x2 matrix would look like this:

\[ 0 = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \]

In the context of this problem, when we compute \(A + (-A)\), each individual element adds up to zero:

\[ \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} + \begin{pmatrix} -a_{11} & -a_{12} \ -a_{21} & -a_{22} \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} \]

Since each element in matrix \(A\) is exactly canceled out by its corresponding negative element in \(-A\), the result is the zero matrix. This proves that \(A + (-A) = 0\).