Given two matrices, $A=\left[\begin{array}{cc}3& 4\\ 2& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}-1& 2\\ 3& -2\end{array}\right]$, find the result of $A+B$.

Step 1 ：Firstly, we consider the respective entries of the matrices. The matrix $A+B$ is obtained by adding the corresponding entries from the matrices A and B.

Step 2 ：The entry in the first row and first column of $A+B$ is obtained by adding the entries in the first row and first column of A and B: $a_{11}+b_{11}=3+(-1)=2$.

Step 3 ：The entry in the first row and second column of $A+B$ is obtained by adding the entries in the first row and second column of A and B: $a_{12}+b_{12}=4+2=6$.

Step 4 ：The entry in the second row and first column of $A+B$ is obtained by adding the entries in the second row and first column of A and B: $a_{21}+b_{21}=2+3=5$.

Step 5 ：The entry in the second row and second column of $A+B$ is obtained by adding the entries in the second row and second column of A and B: $a_{22}+b_{22}=1+(-2)=-1$.

Step 6 ：Therefore, the matrix $A+B$ is given by $\left[\begin{array}{cc}{a}_{11}+b_{11}& a_{12}+b_{12}\\ a_{21}+b_{21}& a_{22}+b_{22}\end{array}\right]=\left[\begin{array}{cc}2& 6\\ 5& -1\end{array}\right]$.