Answer
So, the columns of \( AE \) are:
- Column 1: \( \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} \)
- Column 2: \( \begin{bmatrix} a_{11} + 7a_{22} \\ a_{12} + a_{22} \end{bmatrix} \)
This describes the rows of \( EA \) and the columns of \( AE \) in terms of the elements of matrix \( A \).
Work Step by Step
To describe the rows of \( EA \) and the columns of \( AE \), we need to perform matrix multiplication of matrices \( E \) and \( A \). Given that:
\[ E = \begin{bmatrix} 1 & 7 \\ 0 & 1 \end{bmatrix} \]
And suppose \( A \) is a \( 2 \times n \) matrix (the number of columns is not specified). Let's denote \( A \) as:
\[ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \end{bmatrix} \]
Now, let's compute \( EA \) and \( AE \):
1. \( EA \):
\[ EA = \begin{bmatrix} 1 & 7 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \end{bmatrix} \]
For the first row of \( EA \), we have:
\[ \begin{bmatrix} 1 & 7 \end{bmatrix} \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} = \begin{bmatrix} 1 \cdot a_{11} + 7 \cdot a_{21} \end{bmatrix} \]
For the second row of \( EA \), we have:
\[ \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} = \begin{bmatrix} 0 \cdot a_{11} + 1 \cdot a_{21} \end{bmatrix} \]
So, the rows of \( EA \) are:
- Row 1: \( \begin{bmatrix} a_{11} + 7a_{21} \end{bmatrix} \)
- Row 2: \( \begin{bmatrix} a_{21} \end{bmatrix} \)
2. \( AE \):
\[ AE = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \end{bmatrix} \begin{bmatrix} 1 & 7 \\ 0 & 1 \end{bmatrix} \]
For the first column of \( AE \), we have:
\[ \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a_{11} \cdot 1 \\ a_{21} \cdot 1 \end{bmatrix} \]
For the second column of \( AE \), we have:
\[ \begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix} \begin{bmatrix} 7 \\ 1 \end{bmatrix} = \begin{bmatrix} a_{12} + 7a_{22} \\ a_{12} + a_{22} \end{bmatrix} \]
So, the columns of \( AE \) are:
- Column 1: \( \begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix} \)
- Column 2: \( \begin{bmatrix} a_{11} + 7a_{22} \\ a_{12} + a_{22} \end{bmatrix} \)
This describes the rows of \( EA \) and the columns of \( AE \) in terms of the elements of matrix \( A \).
