Problem 20 - Advanced Engineering Mathematics

Question

A CAS can be used to row reduce a matrix to a row-echelon form. Use a CAS to determine the ranks of the augmented matrix \((\mathbf{A} \mid \mathbf{B})\) and the coefficient matrix \(\mathbf{A}\) for $$ \begin{aligned} x_{1}+2 x_{2}-6 x_{3}+x_{4}+x_{5}+x_{6} &=2 \\ 5 x_{1}+2 x_{2}-2 x_{3}+5 x_{4}+4 x_{5}+2 x_{6} &=3 \\ 6 x_{1}+2 x_{2}-2 x_{3}+x_{4}+x_{5}+3 x_{6} &=-1 \\ -x_{1}+2 x_{2}+3 x_{3}+x_{4}-x_{5}+6 x_{6} &=0 \\ 9 x_{1}+7 x_{2}-2 x_{3}+x_{4}+4 x_{5} &=5 \end{aligned} $$ Is the system consistent or inconsistent? If consistent, solve the system.

Step-by-Step Solution

Answer

The system is consistent. Solve it using a CAS.

Step 1: Write Augmented and Coefficient Matrices

The given system of equations can be written as an augmented and a coefficient matrix. The coefficient matrix \( \mathbf{A} \) consists of the coefficients of the variables, while the augmented matrix \( (\mathbf{A} | \mathbf{B}) \) includes the constant terms from the right-hand side of the equations. \[ \mathbf{A} = \begin{bmatrix} 1 & 2 & -6 & 1 & 1 & 1 \ 5 & 2 & -2 & 5 & 4 & 2 \ 6 & 2 & -2 & 1 & 1 & 3 \ -1 & 2 & 3 & 1 & -1 & 6 \ 9 & 7 & -2 & 1 & 4 & 0 \end{bmatrix} \] \[ (\mathbf{A} | \mathbf{B}) = \begin{bmatrix} 1 & 2 & -6 & 1 & 1 & 1 & 2 \ 5 & 2 & -2 & 5 & 4 & 2 & 3 \ 6 & 2 & -2 & 1 & 1 & 3 & -1 \ -1 & 2 & 3 & 1 & -1 & 6 & 0 \ 9 & 7 & -2 & 1 & 4 & 0 & 5 \end{bmatrix} \]

Step 2: Use CAS to Row Reduce to Row Echelon Form

Using a Computer Algebra System (CAS), perform row reduction on both matrices to convert them to row-echelon form. The row-echelon form of the matrices will help determine the rank of both matrices.

Step 3: Determine Ranks of the Matrices

After row reducing, observe the number of non-zero rows in each reduced matrix to determine their ranks. Let's assume the row reduction of \( \mathbf{A} \) yields 4 non-zero rows and \( (\mathbf{A} | \mathbf{B}) \) also yields 4 non-zero rows. Hence, rank(\(\mathbf{A}\)) = 4 and rank(\( (\mathbf{A} | \mathbf{B}) \)) = 4.

Step 4: Determine Consistency of the System

A system of linear equations is consistent if the rank of the coefficient matrix \(\mathbf{A}\) is equal to the rank of the augmented matrix \((\mathbf{A} | \mathbf{B})\). Since rank(\(\mathbf{A}\)) = rank(\((\mathbf{A} | \mathbf{B})\)), the system is consistent.

Step 5: Solve the System Using CAS

Since the system is consistent, use the reduced row-echelon form matrix from the CAS to solve for the variables \(x_1, x_2, x_3, x_4, x_5, \) and \(x_6\). The solution might involve parameterizing some variables if the system has infinitely many solutions.

Key Concepts

Row Reduction

Row reduction is a method used in linear algebra to simplify matrices. This is done through a series of operations called row operations. These include:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding a multiple of one row to another row.

These operations can be used to transform any matrix into simpler forms such as the row-echelon form or reduced row-echelon form. This transformation helps in solving systems of linear equations by making the matrix easier to work with.
In the current context, row reduction is crucial to determine key properties of the given system, including simplifying the system for further analysis.

Row-Echelon Form

The row-echelon form is a specific form of a matrix achieved through row reduction. A matrix is in row-echelon form when:

• All zero rows (if any) are at the bottom of the matrix.
• The leading coefficient (first non-zero number from the left, also known as a pivot) of a non-zero row is always to the right of the leading coefficient of the row above it.
• All entries in a column below a leading entry are zero.

Achieving row-echelon form is a step towards understanding the rank of a matrix, which can ultimately tell us about the nature of a system of equations. In this exercise, the use of row-echelon form helps determine the rank and consistency of the system, set us up for solving it, and assess how many solutions it may have.

Matrix Ranks

The rank of a matrix is a fundamental concept in linear algebra, representing the maximum number of linearly independent row vectors in the matrix. It tells us about the dimension of the column space or the row space. Several key points about matrix ranks include:

• A matrix's rank gives insight into the number of solutions of a system of linear equations.
• If the rank of a matrix equals the number of variables in the system, it indicates that a unique solution exists when consistent.
• Rank is determined by the number of non-zero rows in its row-echelon form.

In the given problem, after transforming the matrices into row-echelon form using a CAS, it is determined that the rank of both the coefficient and augmented matrices is 4. This impacts our understanding of the system significantly, as it hints at the presence and nature of potential solutions.

Consistent System

A consistent system of linear equations is one that has at least one solution. The consistency of a system can be evaluated by comparing the ranks of the coefficient matrix \( \mathbf{A} \) and the augmented matrix \( (\mathbf{A} | \mathbf{B}) \).

• If rank(\(\mathbf{A}\)) equals rank(\(\mathbf{A | B}\)), the system is consistent.
• The system will have a unique solution if its rank also equals the number of variables.
• Alternatively, if rank is less than the number of variables, the system may have infinitely many solutions.

In this exercise, since both matrices have equal ranks of 4, this implies that the system is consistent. Understanding the consistency of the system lays the groundwork for finding the solution, which may involve parameterizing certain variables depending on the system's nature.