Answer
\( AB = \begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix} \) and \( BA = \begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix} \).
Work Step by Step
To create matrices \( A \) and \( B \), where \( a_{ij} = i + j \) and \( b_{ij} = (-1)^{i+j} \), we first need to specify their dimensions. For this problem, we're creating 2x2 matrices. Let's write them down:
Matrix \( A \):
\[ A = \begin{bmatrix} 1+1 & 1+2 \\ 2+1 & 2+2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} \]
Matrix \( B \):
\[ B = \begin{bmatrix} (-1)^{1+1} & (-1)^{1+2} \\ (-1)^{2+1} & (-1)^{2+2} \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \]
Now, let's multiply these matrices to find \( AB \) and \( BA \).
Matrix multiplication works as follows:
For the product \( AB \), the element in row \( i \) and column \( j \) is computed by taking the dot product of the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \).
For the product \( BA \), the element in row \( i \) and column \( j \) is computed by taking the dot product of the \( i \)-th row of \( B \) and the \( j \)-th column of \( A \).
Let's compute them:
1. \( AB \):
\[ AB = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} = \begin{bmatrix} (2 \cdot 1 + 3 \cdot (-1)) & (2 \cdot (-1) + 3 \cdot 1) \\ (3 \cdot 1 + 4 \cdot (-1)) & (3 \cdot (-1) + 4 \cdot 1) \end{bmatrix} \]
\[ = \begin{bmatrix} (2 - 3) & (-2 + 3) \\ (3 - 4) & (-3 + 4) \end{bmatrix} = \begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix} \]
2. \( BA \):
\[ BA = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} (1 \cdot 2 + (-1) \cdot 3) & (1 \cdot 3 + (-1) \cdot 4) \\ ((-1) \cdot 2 + 1 \cdot 3) & ((-1) \cdot 3 + 1 \cdot 4) \end{bmatrix} \]
\[ = \begin{bmatrix} (2 - 3) & (3 - 4) \\ (-2 + 3) & (-3 + 4) \end{bmatrix} = \begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix} \]
So, \( AB = \begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix} \) and \( BA = \begin{bmatrix} -1 & -1 \\ 1 & 1 \end{bmatrix} \).
