Theory:

Additive Identity:
The null matrix or zero matrix which has all the elements as zero.
 
\(O = \begin{bmatrix}
0 & 0\\
0 & 0
\end{bmatrix}\)
The zero matrix is the identity for matrix addition. When a zero matrix \((O)\) is added to any matrix, say \(A\), the result is always the same matrix \(A\).
 
Let \(A\) be any matrix. Then, \(A+O =O +A = A\).
Example:
Let's take the matrix \( A = \begin{bmatrix}
5 & 10\\ 
4 & 8
\end{bmatrix}\)
 
So, \(A + O = \begin{bmatrix}
5 & 10\\ 
4 & 8
\end{bmatrix} + \begin{bmatrix}
0 & 0\\
0 & 0
\end{bmatrix} = \begin{bmatrix}
5 & 10\\ 
4 & 8
\end{bmatrix}\)
Additive Inverse:
If \(A\) be any given matrix then \(–A\) is the additive inverse of \(A\).
Example:
If \( A = \begin{bmatrix}
5 & -10 & 15\\
6 & 8 & -7\\
-9 & 2 & 14
\end{bmatrix}\) then \( - A = \begin{bmatrix}
-5 & 10 & -15\\
-6 & -8 & 7\\
9 & -2 & -14
\end{bmatrix}\)
Important!
When we add the two additive inverse matrices we get zero matrix
 
That is \(A+(−A) = (−A)+A =O\)