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We are tasked with finding the matrix \( A \) such that:

\[
E_2 E_1 A = I_2
\]

where:

\[
E_1 = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{3} \end{pmatrix}, \quad E_2 = \begin{pmatrix} 1 & -7 \\ 0 & 1 \end{pmatrix}
\]

and \( I_2 \) is the identity matrix:

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

### Step 1: Find \( E_2 E_1 \)

First, multiply \( E_2 \) by \( E_1 \):

\[
E_2 E_1 = \begin{pmatrix} 1 & -7 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{3} \end{pmatrix}
\]

Performing the matrix multiplication:

\[
E_2 E_1 = \begin{pmatrix} 1 \cdot 1 + (-7) \cdot 0 & 1 \cdot 0 + (-7) \cdot \frac{1}{3} \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot \frac{1}{3} \end{pmatrix}
\]

\[
E_2 E_1 = \begin{pmatrix} 1 & -\frac{7}{3} \\ 0 & \frac{1}{3} \end{pmatrix}
\]

### Step 2: Find \( A \)

We need to find \( A \) such that:

\[
\left( \begin{pmatrix} 1 & -\frac{7}{3} \\ 0 & \frac{1}{3} \end{pmatrix} \right) A = I_2
\]

Let \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \). The matrix equation becomes:

\[
\begin{pmatrix} 1 & -\frac{7}{3} \\ 0 & \frac{1}{3} \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

Multiplying these matrices:

\[
\begin{pmatrix} 1 \cdot a + \left(-\frac{7}{3}\right) \cdot c & 1 \cdot b + \left(-\frac{7}{3}\right) \cdot d \\ 0 \cdot a + \frac{1}{3} \cdot c & 0 \cdot b + \frac{1}{3} \cdot d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

This gives the system of equations:

1. \( a - \frac{7}{3}c = 1 \)
2. \( b - \frac{7}{3}d = 0 \)
3. \( \frac{1}{3}c = 0 \)
4. \( \frac{1}{3}d = 1 \)

### Step 3: Solve the system

From equation (3), we get \( c = 0 \).

Substitute \( c = 0 \) into equations (1) and (2):

1. \( a = 1 \)
2. \( b = 0 \)
4. \( d = 3 \)

Thus, the matrix \( A \) is:

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

### Final Answer:

The matrix \( A \) is:

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

Would you like more details or further explanation? Here are some related questions:

1. What is the inverse of an elementary matrix?
2. How do elementary matrices affect solving linear systems?
3. Can this method be applied to larger matrices?
4. What happens if one of the matrices \( E_1 \) or \( E_2 \) is singular?
5. How does matrix multiplication work with non-square matrices?

**Tip:** When solving for matrix \( A \), it's often helpful to think of the process as solving a system of equations for each entry in the matrix.

Suppose that A is a 2x2 matrix such that E2 E1 A = I2. Find the matrix A.

Solve the problem of finding a matrix A such that E2 E1 A = I2, where E1 and E2 are elementary matrices. The solution involves matrix multiplication and solving a system of linear equations. Suitable for students studying Linear Algebra.

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