Question

What is the sum of a matrix $A$ and its negative matrix?

Answer 1

Negative matrix:

The negative matrix of a matrix $A$ is the matrix $\left(-1\right)A$ or $-A$.

The sum of a matrix $A$ and its negative matrix is a null matrix.

For example:

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

The negative matrix is $-A=\left(-1\right)\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$=\left[\begin{array}{cc}-a& -b\\ -c& -d\end{array}\right]$

The sum of matrix $A$ and its negative is $=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]+\left[\begin{array}{cc}-a& -b\\ -c& -d\end{array}\right]$

$$\begin{gathered} =\left[\begin{array}{cc}a-a& b-b\\ c-c& d-d\end{array}\right] \\ =\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ =O \end{gathered}$$

Hence the sum of a matrix $A$ and its negative matrix is a null matrix.