Opposite Matrix

What is the opposite matrix?

The opposite matrix of a given matrix A is defined as a matrix that has the same elements but with opposite signs. It is commonly represented by the symbol -A.

Allow me to provide a practical example.

Consider matrix A, which consists of two rows and three columns.

$$ A = \begin{pmatrix} 1 & -2 & 4 \\ 3 & 0 & 5 \end{pmatrix} $$

The opposite matrix of A, denoted as matrix B, is obtained by negating each element of A.

$$ B = -A = \begin{pmatrix} -1 & 2 & -4 \\ -3 & 0 & -5 \end{pmatrix} $$

In matrix B, each element b(i,j) corresponds to the negated value of element a(i,j) found in matrix A within the same respective row and column.

$$ b_{ij} = -a_{ij} $$

It is important to note that the opposite matrix -A possesses the same dimensions (number of rows and columns) as the original matrix A.

Consequently, it maintains the identical size and numerical composition.

How to calculate the opposite matrix?

To calculate the opposite matrix, a straightforward approach involves multiplying the original matrix by the scalar k=-1.

Let's utilize the aforementioned matrix A as an example:

$$ A = \begin{pmatrix} 1 & -2 & 4 \\ 3 & 0 & 5 \end{pmatrix} $$

By multiplying matrix A with the scalar k=-1, we derive the opposite matrix B.

$$ B = k \cdot A $$

$$ B = -1 \cdot \begin{pmatrix} 1 & -2 & 4 \\ 3 & 0 & 5 \end{pmatrix} $$

$$ B = \begin{pmatrix} -1 \cdot 1 & -1 \cdot (-2) & -1 \cdot 4 \\ -1 \cdot 3 & -1 \cdot 0 & -1 \cdot 5 \end{pmatrix} $$

$$ B = \begin{pmatrix} -1 & 2 & -4 \\ -3 & 0 & -5 \end{pmatrix} $$

Consequently, the resulting matrix B represents the opposite of matrix A, denoted as -A.

$$ B = -A = \begin{pmatrix} -1 & 2 & -4 \\ -3 & 0 & -5 \end{pmatrix} $$

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