Problem 19 - Algebra A Combined Function

Question

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{aligned} 4 x-7 y &=7 \\ 12 x-21 y &=24 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system has no unique solution since the matrix is singular.

Step 1: Write the System as a Matrix Equation

The system \(\begin{aligned} 4x - 7y &= 7 \ 12x - 21y &= 24 \end{aligned}\) can be rearranged in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A = \begin{bmatrix} 4 & -7 \ 12 & -21 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} 7 \ 24 \end{bmatrix}\).

Step 2: Check if the Matrix is Singular

Calculate the determinant of matrix \(A\). \(\det(A) = (4)(-21) - (-7)(12) = -84 + 84 = 0\). Since the determinant of matrix \(A\) is 0, the matrix is singular.

Step 3: Interpret the Singularity

A singular matrix in a system of linear equations indicates that the system does not have a unique solution. In this case, it could either have infinitely many solutions or no solution at all.

Key Concepts

Matrix Equations

Matrix equations provide a structured way of solving systems of linear equations using matrices. They are written in the form:

• The matrix form is expressed as \(A\mathbf{x} = \mathbf{b}\).
• Here, \(A\) is the coefficient matrix containing the coefficients of the variables.
• The vector \(\mathbf{x}\) is a column matrix with the variables of the system.
• The vector \(\mathbf{b}\) is a column matrix with the constants on the right side of the equations.

This approach allows the system to be seen as a single entity, streamlining the process of finding solutions. By manipulating matrices, we can observe relationships and dependencies between the equations, making it easier to apply various mathematical techniques like finding determinants and checking for singularity.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated for a square matrix. It helps determine whether a system of equations has a unique solution. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as:

• \(\det(A) = ad - bc\)

If the determinant is not zero, the matrix is invertible, and the system has a unique solution. However, when the determinant is zero, as in our example, the matrix is singular, meaning it does not have an inverse. This characteristic provides crucial information about the system, such as whether it has infinitely many solutions or no solution at all.

Singular Matrix

When a matrix is singular, its determinant is zero. This indicates a lack of invertibility, which significantly impacts the solutions of the corresponding system of linear equations. A singular matrix implies that the linear equations represented are dependent:

• This dependency means that at least one of the equations can be derived from the others.
• As a result, the system lacks uniqueness in its solutions.

The concept of a singular matrix is crucial for understanding the behavior of linear equations. If the determinant calculation returns zero, it is the first hint that you may need to explore an alternative approach, like checking if there are infinitely many solutions or if the system is inconsistent.

Infinitely Many Solutions

A system of linear equations that has infinitely many solutions usually arises when the matrix is singular. In such a case, the equations are not independent, leading to a situation where multiple solutions coexist:

• There is a dependency among the equations, allowing multiple values for the variables.
• To identify this, you may notice that one equation is simply a multiple of the other.
• Every combination of variables satisfying one equation will satisfy the other.

This redundancy means that instead of pinpointing one intersection point (solution), the solutions form a line or another geometric shape, representing infinite possibilities. Understanding this concept is key to grasping how matrices can reveal the nature of linear equation systems beyond simple numerical solutions.