Given two matrices, \( A = \begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} -1 & 2 \\ 3 & -2 \end{bmatrix} \), find the result of \( A + B \).
Solution
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 \( \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix} = \begin{bmatrix} 2 & 6 \\ 5 & -1 \end{bmatrix} \).
