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The given equation asks for a 2x2 matrix \( A \) such that:

\[
\begin{pmatrix} 5 & 4 \\ 1 & 1 \end{pmatrix} A = \begin{pmatrix} -7 & -1 \\ 0 & -2 \end{pmatrix}.
\]

We can solve for \( A \) by multiplying the inverse of the left matrix \( \begin{pmatrix} 5 & 4 \\ 1 & 1 \end{pmatrix} \) with the right matrix \( \begin{pmatrix} -7 & -1 \\ 0 & -2 \end{pmatrix} \).

The steps involve:

1. **Find the inverse** of the matrix \( \begin{pmatrix} 5 & 4 \\ 1 & 1 \end{pmatrix} \).
2. Multiply the inverse by the right-hand matrix \( \begin{pmatrix} -7 & -1 \\ 0 & -2 \end{pmatrix} \).

Let me calculate the inverse and solve for matrix \( A \).The 2x2 matrix \( A \) that satisfies the equation is:

\[
A = \begin{pmatrix} -7 & 7 \\ 7 & -9 \end{pmatrix}.
\]

Would you like more details on the steps, or have any specific questions?

Here are five related questions for practice:
1. What is the formula to compute the inverse of a 2x2 matrix?
2. How do you check if a matrix is invertible?
3. What happens if the determinant of a matrix is zero?
4. How does matrix multiplication work between two 2x2 matrices?
5. Can you find the product of matrix \( A \) and the matrix \( \begin{pmatrix} -7 & -1 \\ 0 & -2 \end{pmatrix} \)?

**Tip:** Always verify the result by multiplying the original matrix with the inverse to ensure the multiplication yields the identity matrix.

Find a 2x2 matrix A such that \( \begin{pmatrix} 5 & 4 \\ 1 & 1 \end{pmatrix} A = \begin{pmatrix} -7 & -1 \\ 0 & -2 \end{pmatrix} \).

This problem demonstrates how to find a 2x2 matrix A that satisfies the equation \( \begin{pmatrix} 5 & 4 \\ 1 & 1 \end{pmatrix} A = \begin{pmatrix} -7 & -1 \\ 0 & -2 \end{pmatrix} \). Learn the steps of matrix inversion and multiplication with a detailed solution.

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