Math Problem Statement

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

Solution

We are given the matrix AA and its inverse A1A^{-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:

Ax=bA \mathbf{x} = \mathbf{b}

where x=(x1x2x3x4)\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} and b=(01260)\mathbf{b} = \begin{pmatrix} 0 \\ 12 \\ 6 \\ 0 \end{pmatrix}.

To solve for x\mathbf{x}, we can multiply both sides by the inverse matrix A1A^{-1}:

x=A1b\mathbf{x} = A^{-1} \mathbf{b}

Now, we will calculate this multiplication:

\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.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Inverse Matrices
Linear Systems

Formulas

Matrix multiplication A * x = b
Matrix inverse A^{-1} * b = x

Theorems

Matrix Inversion Theorem
Solving Linear Systems using Matrices

Suitable Grade Level

Grades 10-12 (High School Mathematics or Linear Algebra)