Question

Suppose E is the 3 × 3 elementary matrix corresponding to the row operation of switching Row 1 and Row 3, and let A =- 4 & 6 & 2 \\

0 & 1 & 2 \\
5 & -7 & -1
\end{bmatrix} \]
Calculate the product EA, where E is the elementary matrix.
Please provide the step-by-step calculations to find the matrix EA.

Answer 1

To find the product EA for a given elementary matrix E and matrix A, construct E to switch rows 1 and 3, then perform matrix multiplication. The resultant matrix EA will have its first and third rows swapped compared to the original matrix A.

Step-by-step explanation:

To calculate the product EA, where E is the elementary matrix corresponding to switching Row 1 and Row 3, and A is given as
`A = \begin{bmatrix}-4 & 6 & 2 \\ 0 & 1 & 2 \\ 5 & -7 & -1\end{bmatrix}`E. The 3 × 3 elementary matrix E for switching Row 1 and Row 3 is:

`E = \begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix}`

`EA = \begin{bmatrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{bmatrix} \cdot \begin{bmatrix}-4 & 6 & 2 \\ 0 & 1 & 2 \\ 5 & -7 & -1\end{bmatrix} = \begin{bmatrix}5 & -7 & -1 \\ 0 & 1 & 2 \\ -4 & 6 & 2\end{bmatrix}`

Each element of the resulting matrix is found by taking the corresponding row from E and column from A and calculating the sum of the products of the row and column entries. In this case, swapping rows means the first row of A becomes the third row in the result, and the third row of A becomes the first row in the result. The second row remains unchanged since E's middle row is (0 1 0).