Let's first analyze the problem you uploaded. The task is to express the matrix:

$X = \begin{bmatrix} 8 & 1 \\ -1 & 0 \end{bmatrix}$

as a product of elementary matrices:

$X = E_1 E_2 E_3$

To solve this, we need to decompose $X$ into a sequence of elementary row operations represented by matrices $E_1$, $E_2$, and $E_3$.

Step-by-Step Process:

1. Row reduction of $X$: Perform row operations on the matrix $X$ to reduce it to the identity matrix (or a simpler form), noting each row operation as it corresponds to an elementary matrix.
2. Inverse of row operations: After achieving the identity matrix, the reverse of the row operations will give the elementary matrices $E_1$, $E_2$, and $E_3$.

Let me perform the row reduction and compute the corresponding elementary matrices for this process.

1. Row reduction process:

Start with:

$X = \begin{bmatrix} 8 & 1 \\ -1 & 0 \end{bmatrix}$

First step: Use row operations to eliminate the first column, turning the $-1$ in the second row into a 0. We can add $\frac{1}{8}$ times the first row to the second row:

$R_2 \to R_2 + \frac{1}{8} R_1$

This gives the matrix:

$\begin{bmatrix} 8 & 1 \\ 0 & \frac{1}{8} \end{bmatrix}$

The corresponding elementary matrix for this operation is:

$E_1 = \begin{bmatrix} 1 & 0 \\ \frac{1}{8} & 1 \end{bmatrix}$

Second step: Next, make the $\frac{1}{8}$ in the second row, second column equal to 1 by multiplying the second row by 8:

$R_2 \to 8 R_2$

This results in:

$\begin{bmatrix} 8 & 1 \\ 0 & 1 \end{bmatrix}$

The corresponding elementary matrix for this operation is:

$E_2 = \begin{bmatrix} 1 & 0 \\ 0 & 8 \end{bmatrix}$

Third step: To make the element at position $(1,2)$ equal to 0, subtract $1$ times the second row from the first row:

$R_1 \to R_1 - R_2$

This gives:

$\begin{bmatrix} 8 & 0 \\ 0 & 1 \end{bmatrix}$

The corresponding elementary matrix for this operation is:

$E_3 = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$

Final Answer:

The elementary matrices are:

E_2 = \begin{bmatrix} 1 & 0 \\ 0 & 8 \end{bmatrix}, \quad E_3 = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$$ Thus, $$X$$ can be written as the product: $$X = E_1 E_2 E_3$$ Would you like further clarification or any additional details? Here are some related questions: 1. How do we find the inverse of an elementary matrix? 2. Can you describe the relationship between row operations and elementary matrices? 3. What happens if the matrix is not invertible—can we still express it as a product of elementary matrices? 4. How do we perform Gaussian elimination on a 3x3 matrix? 5. What are the properties of elementary matrices? Tip: When performing Gaussian elimination, always check if your row operations are applied correctly to avoid sign errors or incorrect scaling factors.