Problem 2 - College Algebra with Corequisite Support

Question

Examining Cramer's Rule, explain why there is no unique solution to the system when the determinant of your matrix is 0 . For simplicity, use a \(2 \times 2\) matrix.

Step-by-Step Solution

Answer

When the determinant is zero, the system is either dependent or inconsistent, thus no unique solution exists using Cramer's Rule.

Step 1: Define Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns, using determinants. It applies to systems of the form \(AX = B\), where \(A\) is a square matrix, \(X\) is a column vector of variables, and \(B\) is a column vector of constants.

Step 2: Present the Determinant Condition

For Cramer's Rule to provide a unique solution, the determinant of matrix \(A\), denoted as \(|A|\), must be non-zero. If \(|A| = 0\), the system of equations is either dependent (infinitely many solutions) or inconsistent (no solutions), meaning there is no unique solution.

Step 3: Illustrate with a 2x2 Matrix

Consider a generic \(2 \times 2\) matrix \(A\): \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \].\The determinant of \(A\) is calculated as \( |A| = ad - bc \).

Step 4: Show Consequence of Zero Determinant

If \(ad - bc = 0\), then the determinant is zero. In this situation, matrix \(A\) does not have an inverse because a requirement for invertibility is a nonzero determinant. If \(A\) is not invertible, we cannot use Cramer's Rule, as it specifically relies on finding the inverse of \(A\).

Step 5: Connection to Cramer's Rule

With \( |A| = 0 \), Cramer's Rule breaks down since it involves dividing by the determinant to solve for each variable. Since division by zero is undefined, we cannot solve for unique values for \(X\) and \(Y\) in the system.

Key Concepts

Linear Equations

Linear equations are algebraic expressions that represent straight lines in a graph. They involve variables raised to the first power and can be written in the general form:

• ax + by = c

Here, a and b are coefficients, x and y are variables, and c is a constant.
Linear equations can have one or more variables but maintain the characteristic of plotting a straight line when graphed.

• They can appear in many forms, such as slope-intercept and standard form.
• A system of linear equations arises when there are multiple linear equations that share the same set of variables.

In Cramer's Rule, linear equations are solved using determinants, making these equations a cornerstone of the method. Understanding linear equations is critical for grasping more complex concepts like systems of equations and their solution methodologies.

Determinant

A determinant is a special number calculated from a matrix that provides important information about the matrix itself. For a 2x2 matrix, the determinant is computed as:\[ |A| = ad - bc \]where the matrix \( A \) is:\(A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \)The determinant can be single-handedly decisive in determining the properties of the system:

• If a determinant equals zero, the matrix is said to be singular, meaning it doesn’t have an inverse.
• A nonzero determinant indicates that the matrix is invertible and can guarantee a unique solution using methods like Cramer's Rule.

Because the determinant plays such an integral role in solving systems via Cramer's Rule, understanding how to find and interpret it is crucial in linear algebra.

Invertibility

Invertibility refers to a matrix's ability to have an inverse. A matrix, similar to a number, must meet certain conditions to have such a counterpart. The most critical aspect is the determinant:

• A matrix is considered invertible or nonsingular if its determinant is not zero.
• If the determinant is zero, the matrix is singular and cannot be inverted.

An invertible matrix can be thought of as "reversible," meaning multiplying the matrix by its inverse yields the identity matrix:\[ A^{-1} \times A = I \]Here, \( I \) is the identity matrix, operating like the number "1" in standard multiplication. In Cramer's Rule, invertibility is necessary because part of the solution process involves using the inverse of the matrix.

System of Equations

A system of equations consists of multiple equations that are solved together because they share the same variables. For example:\[\begin{align*}ax + by &= e \cx + dy &= f\end{align*}\]This represents two linear equations with variables \(x\) and \(y\). The purpose of solving systems is to find the set of values for the variables that satisfy all equations simultaneously. There are typically three types of outcomes:

• A unique solution occurs if the system is consistent and independent.
• Infinite solutions arise when the system is consistent but dependent.
• No solutions result from inconsistent systems.

Cramer's Rule offers a way to find solutions to systems with a unique answer by using determinants, but it fails if the determinant is zero because the equations are either dependent or there is no solution.