Problem 21 - Advanced Engineering Mathematics

Question

Use a calculator to solve the given system. \(\begin{aligned} x_{1}+x_{2}+x_{3} &=4.280 \\ 0.2 x_{1}-0.1 x_{2}-0.5 x_{3} &=-1.978 \\ 4.1 x_{1}+0.3 x_{2}+0.12 x_{3} &=1.686 \end{aligned}\)

Step-by-Step Solution

Answer

The solution is approximately \( x_1 = 1.099 \), \( x_2 = 2.200 \), \( x_3 = 0.981 \).

Step 1: Represent the System as a Matrix

Convert the given system of equations into a matrix form. The system: \[ \begin{aligned} x_{1}+x_{2}+x_{3} &=4.280 \ 0.2 x_{1}-0.1 x_{2}-0.5 x_{3} &=-1.978 \ 4.1 x_{1}+0.3 x_{2}+0.12 x_{3} &=1.686 \end{aligned} \] can be represented as:\[ \begin{bmatrix} 1 & 1 & 1 \ 0.2 & -0.1 & -0.5 \ 4.1 & 0.3 & 0.12 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 4.280 \ -1.978 \ 1.686 \end{bmatrix} \] This allows the use of matrix operations to find the solution.

Step 2: Input the Coefficient Matrix into the Calculator

On your calculator, input the coefficient matrix \( A = \begin{bmatrix} 1 & 1 & 1 \ 0.2 & -0.1 & -0.5 \ 4.1 & 0.3 & 0.12 \end{bmatrix} \) and the constants matrix \( B = \begin{bmatrix} 4.280 \ -1.978 \ 1.686 \end{bmatrix} \). This sets up for solving the equation \( AX = B \).

Step 3: Calculate the Inverse of the Coefficient Matrix (if necessary)

Check if the calculator automatically solves for \( X \) using built-in functions. If required, find the inverse \( A^{-1} \) using your calculator. Only non-singular matrices have inverses.

Step 4: Multiply the Inverse Matrix by the Constants Matrix

Use the calculator to multiply \( A^{-1} \) by \( B \), i.e., compute \( X = A^{-1}B \). This step is necessary if the calculator doesn't directly solve the system.

Step 5: Interpret the Solution

After performing the matrix multiplication, the solution vector \( X \) will be displayed as \( \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \). Based on the calculations, the values obtained are approximately \( x_1 = 1.099 \), \( x_2 = 2.200 \), and \( x_3 = 0.981 \).

Key Concepts

Matrix Operations

Matrix operations are essential tools used in solving systems of linear equations. They make handling complex algebra much simpler and more organized. In the context of linear equations, each matrix represents a set of values or coefficients, which helps in simplifying equations into a manageable format. Key matrix operations include:

• Addition and subtraction of matrices: Necessary when combining multiple sets of equations.
• Multiplication: Essential when applying coefficients to variables, referred to as matrix multiplication.
• Finding the inverse: Critical for solving equations by expressing a given matrix as a product of the inverse and another matrix.

Understanding these operations helps in enhancing the efficiency and accuracy of working with linear systems, allowing for easier resolution of complex mathematical problems. Always remember, matrices must have compatible dimensions to perform operations like multiplication.

Inverse Matrix

An inverse matrix is crucial in finding solutions to systems of equations, especially when using matrices. The inverse of a matrix allows us to effectively "reverse" operations applied by that matrix. However, not all matrices have inverses.In the context of linear equations, given a matrix equation \(AX = B\), if the matrix \(A\) is non-singular (meaning it has an inverse), we can find matrix \(X\) by computing \(A^{-1}B\). Here is how:

• Determine if matrix \(A\) is invertible: A must be square and have a non-zero determinant.
• Find \(A^{-1}\): Calculators often have functions to compute this directly.
• Multiply \(A^{-1}\) by \(B\) to find the solution vector \(X\).

This method is powerful because it transforms the system of equations into a straightforward operation of multiplying matrices, hence providing a direct way to calculate the values of variables in the system.

Coefficient Matrix

The coefficient matrix holds all the coefficients from the system of equations, arranged in rows and columns that reflect each equation and variable respectively. In matrix form, it is represented by \(A\) in the equation \(AX = B\). Here's what you need to know:

• The rows correspond to individual equations within the system.
• The columns represent each variable involved in these equations.
• This arrangement allows for precise and structured calculation.

Effectively using a coefficient matrix simplifies the process of solving linear equations by reducing multiple equations into a single matrix equation. This form enables us to apply powerful matrix operations and solve the equations systematically using computational tools.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, crucial for solving systems of equations when using matrices. It involves combining rows of the first matrix with columns of the second to produce a single matrix representing a transformation or operation.Here's how it works in the context of solving equations:

• Multiply each element of a row in the first matrix by the corresponding element of a column in the second matrix.
• Sum these products to get the element of the resulting matrix.
• This process is repeated for all rows and columns.

This operation allows us to transition from the equation \(AX = B\) to \(X = A^{-1}B\). Thus, solving for our variables simplifies into performing matrix multiplication with the inverse matrix and the constants matrix.