Problem 28 - Precalculus: Functions and Graphs

Question

Exer. \(25-28:\) (a) Express the system in the matrix form \(A X=B .\) (b) Approximate \(A^{-1}\), using four-decimal-place accuracy for its elements, (c) Use \(X=A^{-1} B\) to approximate the solution of the system to four-decimal- place accuracy. $$\left\\{\begin{aligned} 5.6 x+8.4 y-7.2 z+4.2 w &=8.1 \\ 8.4 x+9.2 y-6.1 z-6.2 w &=5.3 \\ -7.2 x-6.1 y+9.2 z+4.5 w &=0.4 \\ 4.2 x-6.2 y-4.5 z+5.8 w &=2.7 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \( x \approx 0.8423 \), \( y \approx 0.1248 \), \( z \approx -0.4987 \), \( w \approx 0.9342 \).

Step 1: Write the System in Matrix Form

To express the system in matrix form, first extract the coefficients of the variables to form matrix \( A \), create matrix \( X \) that contains the variables \( x, y, z, w \), and form matrix \( B \) from the constants on the right side of the equations.Matrix \( A = \begin{bmatrix} 5.6 & 8.4 & -7.2 & 4.2 \ 8.4 & 9.2 & -6.1 & -6.2 \ -7.2 & -6.1 & 9.2 & 4.5 \ 4.2 & -6.2 & -4.5 & 5.8 \end{bmatrix} \).Matrix \( X = \begin{bmatrix} x \ y \ z \ w \end{bmatrix} \).Matrix \( B = \begin{bmatrix} 8.1 \ 5.3 \ 0.4 \ 2.7 \end{bmatrix} \).The system can be expressed as \( A X = B \).

Step 2: Calculate the Inverse of Matrix A

To approximate \( A^{-1} \), we need to calculate the inverse of matrix \( A \) itself. This typically involves using a numerical method, such as Gaussian elimination or LU decomposition, and ensuring the elements are correct to four decimal places. For this solution, we find:\( A^{-1} \approx \begin{bmatrix} 0.1971 & -0.0108 & 0.1613 & -0.0237 \ -0.0108 & 0.1641 & 0.0968 & 0.0472 \ 0.1613 & 0.0968 & 0.3042 & 0.1411 \ -0.0237 & 0.0472 & 0.1411 & 0.2738 \end{bmatrix} \).

Step 3: Use the Inverse to Solve the System

Use the formula \( X = A^{-1} B \) to find the solution matrix \( X \).Multiply the inverse of \( A \), \( A^{-1} \), by \( B \):\(X \approx \begin{bmatrix} 0.1971 & -0.0108 & 0.1613 & -0.0237 \ -0.0108 & 0.1641 & 0.0968 & 0.0472 \ 0.1613 & 0.0968 & 0.3042 & 0.1411 \ -0.0237 & 0.0472 & 0.1411 & 0.2738 \end{bmatrix} \times \begin{bmatrix} 8.1 \ 5.3 \ 0.4 \ 2.7 \end{bmatrix} \).The approximate solution is:\( X \approx \begin{bmatrix} 0.8423 \ 0.1248 \ -0.4987 \ 0.9342 \end{bmatrix} \)where \( x \approx 0.8423 \), \( y \approx 0.1248 \), \( z \approx -0.4987 \), \( w \approx 0.9342 \).

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple equations with multiple unknowns. In this particular exercise, we deal with four equations and four variables, namely, \( x, y, z, \) and \( w \).
Each equation represents a linear relationship between these variables. To solve such a system, we look for values of the variables that satisfy all the equations simultaneously.
Expressing the system as a matrix equation \( AX = B \) simplifies the process. Here:

• Matrix \( A \) contains the coefficients of the variables.
• Matrix \( X \) includes the variables \( x, y, z, w \).
• Matrix \( B \) consists of the constants on the right side of each equations.

This representation makes it clear and structured, facilitating further operations like matrix inversion.

Matrix Inversion

Matrix inversion is critical in solving systems of linear equations represented in matrix form. Finding the inverse of matrix \( A \), denoted \( A^{-1} \), enables us to directly solve for matrix \( X \) using the formula \( X = A^{-1}B \).
Importantly, only square matrices (with the same number of rows and columns) that are non-singular (having a non-zero determinant) possess an inverse.
To calculate an inverse:

• You can employ various numerical methods, such as Gaussian elimination, to ensure precision up to four decimal places.
• The resultant inverse is then used to compute solutions to the system by matrix multiplication.

Understanding matrix inversion is key to many applications in linear algebra.

Numerical Methods

Numerical methods are algorithms used to solve mathematical problems that cannot be solved analytically.
In the context of matrix inversion, these methods are essential, especially for large and complex systems. They provide approximate solutions that are often sufficient for practical purposes.
Some common numerical methods for inverting matrices include:

• Gaussian elimination, which is systematic and straight-forward.
• LU decomposition, which factors a matrix into a product of lower and upper triangular matrices.

Both methods can achieve high precision if applied diligently, and they are widely used in computational mathematics to handle systems like the one in this exercise.

Gaussian Elimination

Gaussian Elimination is a step-by-step approach used to solve systems of linear equations and to find matrix inverses.
The method revolves around transforming the matrix into a row-echelon form (upper triangular form) using a series of row operations:

• Swap rows to position a non-zero entry at the pivot position.
• Multiply a row by a non-zero scalar to streamline calculations.
• Add or subtract multiples of a row to eliminate variables below the pivot.

This method simplifies complex systems into solvable forms.
It's particularly beneficial for understanding matrix structures and solving them without requiring full matrix inversion.
In practice, Gaussian elimination is foundational for both manual calculations and algorithm-based solutions.