https://file.aiquickdraw.com/mathbot/file/1728661100846n5isyflg.jpg

Let's solve question 7 from the image, which involves solving the system of matrix equations given as:

\[
\begin{bmatrix} 
1 & 4 \\
-1 & 7 \\
4 & -4 
\end{bmatrix} x_1
+
\begin{bmatrix} 
-5 & 3 \\
3 & -5 \\
1 & 2 
\end{bmatrix} x_2
+
\begin{bmatrix} 
7 & -8 \\
0 & 2 \\
0 & -7 
\end{bmatrix} x_3
=
\begin{bmatrix} 
6 \\
0 \\
-7 
\end{bmatrix}
\]

We need to solve this system for \(x_1\), \(x_2\), and \(x_3\). Let me work this out.It seems there was an issue processing the matrix system. Let's solve it manually instead:

The given system of equations is:

\[
\begin{aligned}
    A_1x_1 + A_2x_2 + A_3x_3 &= B
\end{aligned}
\]

Where:

\[
A_1 = \begin{bmatrix} 1 & 4 \\ -1 & 7 \\ 4 & -4 \end{bmatrix}, \quad
A_2 = \begin{bmatrix} -5 & 3 \\ 3 & -5 \\ 1 & 2 \end{bmatrix}, \quad
A_3 = \begin{bmatrix} 7 & -8 \\ 0 & 2 \\ 0 & -7 \end{bmatrix}, \quad
B = \begin{bmatrix} 6 \\ 0 \\ -7 \end{bmatrix}
\]

To solve, we can break it down row by row and solve for the unknowns \(x_1\), \(x_2\), and \(x_3\). I'll solve this system now.The solution to the given system of equations is:

\[
x_1 = -\frac{86}{111}, \quad x_2 = \frac{17}{37}, \quad x_3 = \frac{35}{111}
\]

Would you like further clarification on any steps? Here are some related questions:

1. How can matrix equations be solved using Gaussian elimination?
2. What are the conditions for a matrix equation \(Ax = b\) to have a unique solution?
3. How do we interpret the determinant of a matrix in relation to solving systems?
4. What is the significance of matrix inverses in solving linear systems?
5. Can non-square matrices have unique solutions in matrix equations?

**Tip:** Always check the determinant of the coefficient matrix to verify if the system is invertible.

Solve the matrix equation involving x1, x2, and x3 as shown in the image.

Explore the step-by-step solution to the system of matrix equations involving variables x1, x2, and x3. This problem covers key linear algebra concepts, including matrix multiplication and solving systems of equations. Ideal for undergraduate students.

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