Problem 43 - Precalculus: Mathematics for Calculus

Question

Solving a Linear System as a Matrix Equation Solve the system of equations by converting to a matrix equation and using the inverse of the coefficient matrix, as in Example \(6 .\) $$\left\\{\begin{aligned}2 x+4 y+z &=7 \\\\-x+y-z &=0 \\\x+4 y &=-2\end{aligned}\right.$$

Step-by-Step Solution

Answer

Find \( A^{-1} \), solve \( \mathbf{x} = A^{-1} \mathbf{b} \), and verify solutions.

Step 1: Write the system in matrix form

Begin by expressing the system of linear equations in the matrix form \( A \mathbf{x} = \mathbf{b} \). Extract coefficients for the matrix \( A \), the variables \( x \), and the constants for \( \mathbf{b} \):\[A = \begin{bmatrix} 2 & 4 & 1 \ -1 & 1 & -1 \ 1 & 4 & 0 \end{bmatrix}, \;\mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \;\mathbf{b} = \begin{bmatrix} 7 \ 0 \ -2 \end{bmatrix}\]

Step 2: Find the inverse of the coefficient matrix

Calculate the inverse of matrix \( A \), denoted \( A^{-1} \). Use the formula for the inverse of a 3x3 matrix, or other computational techniques like row reduction or a calculator capable of matrix operations.

Step 3: Solve for the variable matrix

Multiply the inverse of the coefficient matrix \( A^{-1} \) by the vector \( \mathbf{b} \) to solve for the variable matrix \( \mathbf{x} \):\[ \mathbf{x} = A^{-1} \mathbf{b} \]

Step 4: Calculate specific solutions

Execute the matrix multiplication to find the values of the variables. Ensure the calculations are correct by multiplying:\[ \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix} = A^{-1} \mathbf{b} \]

Step 5: Verify the solution

Substitute the values of \( x \), \( y \), and \( z \) back into the original equations to confirm they satisfy all the equations. This step helps verify the solution's accuracy.

Key Concepts

Linear Systems

In the world of mathematics, a linear system consists of two or more linear equations involving the same set of variables. Solving these systems is crucial as it helps in finding points of intersection, optimal solutions in operations research, and is widely used in engineering problems. A linear system can often be solved by expressing it in terms of matrices, which makes complex computations much more manageable.

When dealing with linear systems:

• Each equation gives a line, plane, or hyperplane, depending on the number of variables and equations.
• The solution(s) to the system is the point(s) where these geometrical shapes intersect.
• For a practical system of linear equations, we usually seek a common solution for all equations, usually represented as a set of values for the involved variables.

Linear systems can be represented in matrix notation as \( A \mathbf{x} = \mathbf{b} \), where:

• \( A \) is the coefficient matrix composed of all the coefficients in the system.
• \( \mathbf{x} \) is the variable vector.
• \( \mathbf{b} \) is the constant vector containing the results of the equations.

The transition from equations to matrices allows for the usage of matrix operations in solving the system, making it convenient and sometimes imperative in computational algorithms.

Matrix Inversion

The concept of matrix inversion plays a critical role when solving linear systems using matrices. The inverse of a matrix \( A \), denoted as \( A^{-1} \), is a crucial tool because it allows us to solve for \( \mathbf{x} \) in the equation \( A \mathbf{x} = \mathbf{b} \). To find \( \mathbf{x} \), one can rearrange the equation as \( \mathbf{x} = A^{-1} \mathbf{b} \).

Matrix inversion involves:

• Ensuring the matrix \( A \) is square (same number of rows and columns) and nonsingular (determinant is not zero).
• Utilizing techniques like the adjugate method, Gauss-Jordan elimination, or computational tools to find \( A^{-1} \).
• Ensuring accurate computation since the inverse allows back-solving the initial equation for solutions.

Remember, not every matrix is invertible. This means, for some linear systems represented as \( A \mathbf{x} = \mathbf{b} \), the coefficient matrix \( A \) might not have an inverse, translating to systems with no unique solutions or having infinitely many.

Solution Verification

After obtaining a solution from a linear system, it is critical to verify its accuracy. This step ensures that no computational errors occurred during matrix operations and that the final values satisfy all original equations.

Verification involves:

• Substituting the calculated values for \( x, y, \) and \( z \) back into each equation of the original linear system.
• Ensuring that each side of the equations balances correctly.
• Checking for consistent results across all equations, indicative of a correct solution.

This straightforward step allows identifying errors in more complex calculations where an oversight may lead to incorrect interpretations of the results. Verification also builds confidence in the solution, especially when sharing outcomes in collaborative and academic settings.