WRITE X = "-3" 1, "-8" 0 AS A PRODUCT OF X = E1 E2 E3 OF ELEMENTARY MATRICES. WHAT IS E1 , E2 AND E3

To write the matrix \( X = \begin{pmatrix} -3 & 1 \\ -8 & 0 \end{pmatrix} \) as a product of elementary matrices \( E_1, E_2, E_3 \), we need to understand how we can transform the identity matrix into \( X \) using elementary row operations. Each elementary row operation corresponds to multiplying the matrix by an elementary matrix from the left.

Let’s go through the steps to transform the identity matrix \( I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \) into \( X \), step by step, recording each elementary matrix.

### Step 1: Scale the first row by -3
We want to get the first element of the first row to -3. To do this, we can multiply the first row by -3. The elementary matrix \( E_1 \) that performs this operation is:

\[
E_1 = \begin{pmatrix} -3 & 0 \\ 0 & 1 \end{pmatrix}
\]

When we multiply \( E_1 \) by \( I_2 \), we get:

\[
E_1 I_2 = \begin{pmatrix} -3 & 0 \\ 0 & 1 \end{pmatrix}
\]

### Step 2: Add 8 times the first row to the second row
Now, we want to get the \(-8\) in the second row, first column. To do this, we can add 8 times the first row to the second row. The elementary matrix \( E_2 \) for this row operation is:

\[
E_2 = \begin{pmatrix} 1 & 0 \\ 8 & 1 \end{pmatrix}
\]

When we multiply \( E_2 \) by the matrix from Step 1, we get:

\[
E_2 \begin{pmatrix} -3 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -3 & 0 \\ -8 & 1 \end{pmatrix}
\]

### Step 3: Set the second row, second column to 0
Finally, we need to make the second row, second column equal to 0. To do this, we can subtract the second column from the first column, which corresponds to the elementary matrix \( E_3 \):

\[
E_3 = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}
\]

However, it seems there's a misunderstanding

Explore how to express the matrix X = [[-3, 1], [-8, 0]] as a product of elementary matrices E1, E2, and E3. This solution covers elementary row operations and matrix transformations, essential for linear algebra students. Suitable for undergraduate-level learners.

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