https://file.aiquickdraw.com/mathbot/file/172772769168777aul37z.jpg

We are given the matrix \( A \) and its inverse \( A^{-1} \), along with a system of equations to solve. The key to solving this system is utilizing the inverse matrix.

The system of equations can be written in matrix form as:

\[
A \mathbf{x} = \mathbf{b}
\]

where \( \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 0 \\ 12 \\ 6 \\ 0 \end{pmatrix} \).

To solve for \( \mathbf{x} \), we can multiply both sides by the inverse matrix \( A^{-1} \):

\[
\mathbf{x} = A^{-1} \mathbf{b}
\]

Now, we will calculate this multiplication:

\[
\mathbf{x} = \begin{pmatrix}
\frac{2}{3} & -\frac{2}{3} & -\frac{1}{2} & -\frac{13}{6} \\
\frac{1}{3} & \frac{1}{6} & -\frac{1}{4} & -\frac{13}{12} \\
-1 & 1 & 1 & 3 \\
\frac{2}{3} & -\frac{2}{3} & -\frac{1}{2} & -\frac{7}{6}
\end{pmatrix}
\begin{pmatrix} 0 \\ 12 \\ 6 \\ 0 \end{pmatrix}
\]

Multiplying this out:

1. First row:
\[
x_1 = \frac{2}{3}(0) + \left(-\frac{2}{3}\right)(12) + \left(-\frac{1}{2}\right)(6) + \left(-\frac{13}{6}\right)(0) = -8 - 3 = -11
\]

2. Second row:
\[
x_2 = \frac{1}{3}(0) + \frac{1}{6}(12) + \left(-\frac{1}{4}\right)(6) + \left(-\frac{13}{12}\right)(0) = 2 - 1.5 = 0.5
\]

3. Third row:
\[
x_3 = -1(0) + 1(12) + 1(6) + 3(0) = 12 + 6 = 18
\]

4. Fourth row:
\[
x_4 = \frac{2}{3}(0) + \left(-\frac{2}{3}\right)(12) + \left(-\frac{1}{2}\right)(6) + \left(-\frac{7}{6}\right)(0) = -8 - 3 = -11
\]

So, the solution vector is:

\[
\mathbf{x} = \begin{pmatrix} -11 \\ 0.5 \\ 18 \\ -11 \end{pmatrix}
\]

Therefore, the value of \( x_4 \) is \( -11 \).

Would you like more details on how the matrix multiplication was performed or any other clarification?

Here are five related questions to expand on this topic:
1. How do you compute the inverse of a matrix manually?
2. What is the significance of matrix inversion in solving linear systems?
3. Can all matrices be inverted, and why might a matrix not have an inverse?
4. What is the geometric interpretation of matrix multiplication in this context?
5. How would the solution change if the matrix was not invertible?

**Tip**: Always check if a matrix is invertible by computing its determinant before attempting to solve a system using the inverse.

Make use of the inverse matrix to solve the system of equations and find the value of x_4.

Learn how to solve a system of equations using the inverse of a given matrix. This problem involves calculating the inverse matrix and solving the system using matrix multiplication. The solution is detailed for high school students or early linear algebra learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation