
Consider the following matrices.

A = 	
1	−2
−4	6
,  B = 	
1	−2
0	−2

Find an elementary matrix E such that EA = B

To solve for the elementary matrix \( E \) such that \( EA = B \), we need to determine what row operations were applied to matrix \( A \) to transform it into matrix \( B \). Elementary matrices correspond to these row operations.

### Matrices A and B:

\[
A = \begin{pmatrix} 
1 & -2 \\
-4 & 6 
\end{pmatrix}, \quad B = \begin{pmatrix} 
1 & -2 \\
0 & -2 
\end{pmatrix}
\]

#### Step 1: Identify row operations

We can see that matrix \( B \) is obtained by applying the following row operation to \( A \):

- **Row 2** in matrix \( B \) is a result of adding 4 times row 1 to row 2 in matrix \( A \).

So the row operation is:
\[
R_2 \rightarrow R_2 + 4R_1
\]

#### Step 2: Express the row operation as an elementary matrix

The elementary matrix corresponding to the row operation \( R_2 \rightarrow R_2 + 4R_1 \) is:

\[
E = \begin{pmatrix} 
1 & 0 \\
4 & 1 
\end{pmatrix}
\]

#### Step 3: Verify \( E \times A = B \)

Now, let's verify that multiplying \( E \) with \( A \) results in \( B \).

\[
EA = \begin{pmatrix} 
1 & 0 \\
4 & 1 
\end{pmatrix} 
\begin{pmatrix} 
1 & -2 \\
-4 & 6 
\end{pmatrix} 
= \begin{pmatrix} 
(1)(1) + (0)(-4) & (1)(-2) + (0)(6) \\
(4)(1) + (1)(-4) & (4)(-2) + (1)(6) 
\end{pmatrix}
= \begin{pmatrix} 
1 & -2 \\
0 & -2 
\end{pmatrix}
\]

This is exactly matrix \( B \).

Thus, the elementary matrix \( E \) is:

\[
E = \begin{pmatrix} 
1 & 0 \\
4 & 1 
\end{pmatrix}
\]

Would you like further details on elementary matrices or any other clarifications?

Here are 5 related questions to expand on this topic:
1. How do elementary matrices relate to row operations?
2. Can every matrix be transformed into another by an elementary matrix?
3. What are the different types of elementary row operations?
4. How would you compute the inverse of an elementary matrix?
5. What happens when multiple elementary matrices are applied successively?

**Tip**: When multiplying matrices, always ensure that the dimensions are compatible: the number of columns in the first matrix must match the number of rows in the second matrix.

Consider the following matrices.
A = 1 −2
       −4 6,
B = 1 −2
      0 −2.
Find an elementary matrix E such that EA = B.

Learn how to find the elementary matrix E that transforms matrix A to matrix B using row operations. This linear algebra problem focuses on elementary matrices, matrix multiplication, and row operations. Suitable for undergraduate students in a linear algebra course.

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