The problem asks for the sum of a matrix $A$ and its negative $-A$. The matrix $A$ is given as: $A = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 1 & 2 \\ 5 & 3 & 0 \end{bmatrix}$

1. Problem Description

The problem asks for the sum of a matrix $A$ and its negative $-A$. The matrix $A$ is given as:

$A = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 1 & 2 \\ 5 & 3 & 0 \end{bmatrix}$

2. Solution Steps

Let $A$ be a matrix. The negative of matrix $A$, denoted by $-A$, is obtained by multiplying each element of $A$ by $-1$.

So, if $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$, then $-A = \begin{bmatrix} -a & -b & -c \\ -d & -e & -f \\ -g & -h & -i \end{bmatrix}$.

Given $A = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 1 & 2 \\ 5 & 3 & 0 \end{bmatrix}$, then $-A = \begin{bmatrix} -2 & 0 & -1 \\ -3 & -1 & -2 \\ -5 & -3 & 0 \end{bmatrix}$.

We want to find the sum of $A$ and $-A$, which is $A + (-A)$.

$A + (-A) = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 1 & 2 \\ 5 & 3 & 0 \end{bmatrix} + \begin{bmatrix} -2 & 0 & -1 \\ -3 & -1 & -2 \\ -5 & -3 & 0 \end{bmatrix} = \begin{bmatrix} 2+(-2) & 0+0 & 1+(-1) \\ 3+(-3) & 1+(-1) & 2+(-2) \\ 5+(-5) & 3+(-3) & 0+0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$.

Therefore, the sum of a matrix and its negative is the zero matrix.

3. Final Answer

$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$