Problem 11 - Differential Equations and Linear Algebra

Question

},\( of the given system. $$\begin{aligned}x_{1}+2 x_{2}-… # Determine the coefficient matrix, \)A\(, the right-hand side vector, \)\mathbf{b},\( and the augmented matrix, \)A^{\\#},$ of the given system. $$\begin{aligned}x_{1}+2 x_{2}-x_{3} &=0 \\\2 x_{1}+3 x_{2}-2 x_{3} &=0 \\\5 x_{1}+6 x_{2}-5 x_{3} &=0\end{aligned}$$

Step-by-Step Solution

Answer

The coefficient matrix A, right-hand side vector b, and the augmented matrix A∼ are: \[A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 3 & -2 \\ 5 & 6 & -5 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \quad A^{\\#} = \begin{bmatrix} 1 & 2 & -1 & | & 0 \\ 2 & 3 & -2 & | & 0 \\ 5 & 6 & -5 & | & 0 \end{bmatrix}.\]

Step 1: Form the Coefficient Matrix A

To form the coefficient matrix A, write the coefficients of the variables from the given system of linear equations. \[A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 3 & -2 \\ 5 & 6 & -5 \end{bmatrix}\]

Step 2: Form the Right-Hand Side Vector b

The right-hand side vector b consists of the constants on the other side of the equal sign in the system of linear equations. In our case, they are all-zero. \[\mathbf{b} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\]

Step 3: Form the Augmented Matrix A∼

To form the augmented matrix A∼, just add the right-hand side vector b as the last column to the matrix A. \[A^{\\#} = \begin{bmatrix} 1 & 2 & -1 & | & 0 \\ 2 & 3 & -2 & | & 0 \\ 5 & 6 & -5 & | & 0 \end{bmatrix}\] So, the coefficient matrix A, right-hand side vector b, and the augmented matrix A∼ are: \[A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 3 & -2 \\ 5 & 6 & -5 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, \quad A^{\\#} = \begin{bmatrix} 1 & 2 & -1 & | & 0 \\ 2 & 3 & -2 & | & 0 \\ 5 & 6 & -5 & | & 0 \end{bmatrix}.\]

Key Concepts

Coefficient Matrix

A coefficient matrix is an essential part in solving systems of linear equations. It is formed by extracting just the coefficients of the variables in the system. For the system of equations:\[\begin{aligned}x_{1}+2x_{2}-x_{3} &= 0, \2x_{1}+3x_{2}-2x_{3} &= 0, \5x_{1}+6x_{2}-5x_{3} &= 0\end{aligned}\]
the coefficients of the variables \(x_1, x_2,\) and \(x_3\) are listed into a matrix.
Here, our coefficient matrix looks like this:\[A = \begin{bmatrix} 1 & 2 & -1 \ 2 & 3 & -2 \ 5 & 6 & -5 \end{bmatrix}\]

• The entries in the matrix correspond to the coefficients of each variable across the three equations.
• Each row represents the coefficients from one equation.
• This matrix helps in identifying the systematic arrangement of the equations' coefficients.

Understanding how to extract and write the coefficient matrix is crucial in the techniques for solving linear equations like matrix inversion or Gaussian elimination.

Augmented Matrix

An augmented matrix combines the coefficient matrix with the right-hand side of the equations. This matrix representation is used to simplify solving the system of linear equations through methods like Gaussian elimination. From the given system of equations, we already have the coefficient matrix \( A \) as:\[\begin{bmatrix} 1 & 2 & -1 \ 2 & 3 & -2 \ 5 & 6 & -5 \end{bmatrix}\]And the right-hand side vector \( \mathbf{b} \) which consists entirely of zeros:\[\begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}\]The augmented matrix \( A^{\#} \), therefore, combines these two into one matrix by appending the right-hand side vector as an additional column:\[\begin{bmatrix} 1 & 2 & -1 & | & 0 \ 2 & 3 & -2 & | & 0 \ 5 & 6 & -5 & | & 0 \end{bmatrix}\]

• The vertical line is often used in augmented matrices to denote the boundary between the coefficient matrix and the right-hand side.
• It provides a visual aid to separate the coefficients from constants, making operations on the matrix more intuitive.
• This form aids directly in processes like row reduction to solve the equations.

By understanding the augmented matrix, you can efficiently apply techniques for finding solutions to linear systems.

Right-Hand Side Vector

The right-hand side (RHS) vector is an integral part of expressing a system of equations in matrix form. For the system you're working with:\[\begin{aligned}x_{1}+2x_{2}-x_{3} &= 0, \2x_{1}+3x_{2}-2x_{3} &= 0, \5x_{1}+6x_{2}-5x_{3} &= 0\end{aligned}\]The RHS vector stems from the constants found on the right side of each equation. In this instance, every equation is equal to zero, so our RHS vector becomes:\[\mathbf{b} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}\]

• Each element in the RHS vector represents the constant term to which the left-hand side of the corresponding equation equates.
• For homogeneous systems of linear equations, like this one, the RHS vector will always consist of zeros.
• This vector is crucial in forming the augmented matrix as it transitions the set of equations into a single entity which can be manipulated for solutions.

Understanding the RHS vector is fundamental in setting up and solving systems of equations, whether they are homogenous or not.