Question

Use determinants to show that the following system is inconsistent. $$ \begin{aligned} &\mathrm{x}+\mathrm{y}=3(\mathrm{x}-2 \mathrm{y})+5 \\ &14 \mathrm{y}-4 \mathrm{x}=11 \end{aligned} $$

Short answer

The given system of linear equations is inconsistent. We rewrote the system in standard form, formed the coefficient matrix and the augmented matrix, and calculated their determinants. The coefficient matrix had a non-zero determinant of 34, while the augmented matrix had an undefined determinant. This indicates an inconsistent system with no solutions.

Step-by-step solution

1. Rewrite the system in standard form

First, we'll rewrite the given system of linear equations in standard form: $$ \begin{aligned} x+y &= 3(x-2y)+5 \\ 14y-4x &= 11 \end{aligned} $$ Standard form is obtained by expanding and regrouping the terms: $$ \begin{aligned} 1x-5y &= 5 \\ 4x+14y &= 11 \end{aligned} $$

2. Form the coefficient matrix and the augmented matrix

Now, let's create the coefficient matrix and the augmented matrix for the given system. The coefficient matrix contains the coefficients of x and y, while the augmented matrix includes an additional column for the constants. Coefficient matrix: $$ \begin{bmatrix} 1 & -5 \\ 4 & 14 \end{bmatrix} $$ Augmented matrix: $$ \begin{bmatrix} 1 & -5 & 5 \\ 4 & 14 & 11 \end{bmatrix} $$

3. Calculate the determinants

Next, we'll calculate the determinant of the coefficient matrix and the augmented matrix. Determinant of the coefficient matrix: $$ \det\begin{bmatrix} 1 & -5 \\ 4 & 14 \end{bmatrix} = 1(14) - 4(-5) = 14 + 20 = 34 $$ Determinant of the augmented matrix: $$ \det\begin{bmatrix} 1 & -5 & 5 \\ 4 & 14 & 11 \\ 0 & 0 & 1 \end{bmatrix} $$ Since the third row has two zero entries except for the last entry, the determinant is not defined.

4. Analyze the determinants for inconsistency

The determinant of the coefficient matrix, \(34\), is non-zero, so it means the system is either consistent with a unique solution or inconsistent. However, the determinant of the augmented matrix is not defined, which implies that the system is inconsistent and has no solutions. The given system of linear equations is inconsistent as proven by the determinants method.

Key concepts

Determinants

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix that can include whether the system of equations related to it has a unique solution or not. In simple terms, determinants can tell us if the system is consistent (having at least one or exactly one solution) or inconsistent (having no solution).

When dealing with a 2x2 matrix, as in the given exercise, the determinant is calculated by crossing multiplying the diagonals and subtracting one product from the other. In formulaic terms, for a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( \text{det}(A) = ad - bc \).

If the determinant of a coefficient matrix in a linear system is zero, it implies that the system may be dependent (having infinitely many solutions) or inconsistent. If non-zero, as in our exercise (34), it usually means the system is independent and could be consistent with a unique solution. However, as we will discuss in the next sections, other factors contribute to determining consistency.

Coefficient Matrix

The coefficient matrix is a matrix consisting strictly of the coefficients of the variables in a system of linear equations. It's a square matrix if the number of equations is equal to the number of variables. In our system, the coefficient matrix is derived from the left side of the equations, ignoring the constants (right side of the equations).

This matrix plays an essential role in solving systems of equations. By observing the coefficient matrix, we can apply certain operations to determine the solvability of the system without altering the solutions. These operations include row reduction techniques to achieve reduced row echelon form or calculating determinants.

In this exercise, the coefficient matrix is \(\begin{bmatrix}1 & -5 \ 4 & 14\end{bmatrix}\). When we look at the coefficient matrix separately from the constants, we can analyze the potential relationship between the variables x and y. Since the determinant of this matrix is 34, a non-zero value, it initially suggests the system may have a unique solution.

Augmented Matrix

An augmented matrix is an extension of the coefficient matrix that includes an additional column for the constant terms from each linear equation in the system. It's a convenient way to organize all the information from a system of linear equations for procedures like row reduction or determinant calculation.

Though the operations performed on the augmented matrix do not affect the solvability of the system, having an inconsistent third row after such operations, as seen in this exercise, does hint at inconsistency. The third row with zeros in the coefficient section and a non-zero value in the constants section implies that the system cannot satisfy all equations simultaneously.

Using the determinant of the augmented matrix can also be a tool to determine the consistency of the system. However, in this exercise, we noticed that the determinant of the augmented matrix is not defined, which is in agreement with the presence of an inconsistent row. Thus, it confirms that the system does not have a solution - it's inconsistent.