Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by using the matrix inverse method. $$\begin{aligned} x+3 y-2 z-w &=9 \\ 4 x+y+z+2 w &=2 \\ -3 x-y+z-w &=-5 \\ x-y-3 z-2 w &=2 \end{aligned}$$

Step-by-Step Solution

Answer

The solution vector is \( \begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix} \).

Step 1: Write the System of Equations in Matrix Form

The given system of equations can be written in matrix form as \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the variable matrix, and \( \mathbf{b} \) is the constant matrix. For the given system:\[A = \begin{pmatrix} 1 & 3 & -2 & -1 \ 4 & 1 & 1 & 2 \ -3 & -1 & 1 & -1 \ 1 & -1 & -3 & -2 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \ y \ z \ w \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 9 \ 2 \ -5 \ 2 \end{pmatrix}\]

Step 2: Find the Inverse of the Coefficient Matrix

To solve the system \( A\mathbf{x} = \mathbf{b} \) using the matrix inverse method, we need to find the inverse of the matrix \( A \). Compute the inverse, \( A^{-1} \), if it exists.

Step 3: Multiply the Inverse by the Constant Matrix

Once the inverse matrix \( A^{-1} \) is found, calculate \( \mathbf{x} = A^{-1} \mathbf{b} \). This step involves multiplying the inverse matrix by the constant matrix \( \mathbf{b} \) to find the solution for \( \mathbf{x} \).

Step 4: Verify the Solution

After obtaining \( \mathbf{x} \), substitute the values of \( x, y, z, \) and \( w \) back into the original equations to verify that they satisfy all equations. This ensures the solution is correct.

Key Concepts

System of Equations

A system of equations is a set of equations all at once. These equations are related and share the same set of variables. In the exercise, we have four equations with the variables \( x, y, z, \) and \( w \). Each equation represents a line (or hyperplane in higher dimensions) and solving them involves finding a common point or set of points that satisfies all equations at the same time. This is like finding where all the lines intersect in a graphical sense. In the algebraic context, our goal is to find unique values for \( x, y, z, \) and \( w \) that satisfy all these equations simultaneously. This can often be done using methods that allow us to handle multiple equations at once, such as the matrix inverse method.

Coefficient Matrix

A coefficient matrix, denoted as \( A \) in our exercise, is a way to neatly organize and represent the coefficients of the variables from all equations in the system. This forms a matrix, which is simply a rectangular array of numbers.
For example, in the given system of equations, the coefficient matrix is:\[ A = \begin{pmatrix} 1 & 3 & -2 & -1 \ 4 & 1 & 1 & 2 \ -3 & -1 & 1 & -1 \ 1 & -1 & -3 & -2 \end{pmatrix} \]
This matrix contains all the numbers multiplying the variables \( x, y, z, \) and \( w \) from each equation. By organizing this information into a matrix, it becomes easier to apply mathematical operations, such as finding its inverse, which are helpful in solving the system.

Matrix Multiplication

Matrix multiplication is an operation between two matrices where the rows of the first matrix are multiplied by the columns of the second matrix, resulting in a new matrix. This process is crucial when using the matrix inverse method to solve a system of equations.
In our exercise, once we find the inverse of the coefficient matrix \( A \), denoted \( A^{-1} \), we multiply it by the constant matrix \( \mathbf{b} \). This is done using matrix multiplication to find the solution matrix \( \mathbf{x} \):\[ \mathbf{x} = A^{-1} \mathbf{b} \]
The calculations behind matrix multiplication can seem complex, but they efficiently enable us to solve for the variables when direct computation from the system appears challenging.

Solution Verification

After obtaining potential solutions for \( x, y, z, \) and \( w \) through methods like matrix inversion, it's essential to check if these values are correct. Solution verification involves substituting these values back into the original set of equations to ensure each equation is satisfied.
This helps confirm that the calculated solution genuinely works for every equation in the system. If, after substitution, all equations hold true, the solution is verified and considered correct. But if one or more equations don't satisfy, it suggests an error was made during calculations, or the solution may be incorrect or incomplete. Hence, verification is a crucial step in concluding our problem-solving process.