Problem 71 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 71-74, determine whether the statement is true or false. Justify your answer. In Cramer's Rule, the numerator is the determinant of the coefficient matrix.

Step-by-Step Solution

Answer

False

Step 1: Understanding Cramer's Rule

Cramer's Rule is a formula used to solve systems of linear equations by expressing the solution in terms of determinants. It is defined as follows: Given a system of linear equations, the solution for any variable say \(x_i\) can be given as \[x_i = \frac{D_i}{D}\] where \(D\) is the determinant of the coefficient matrix and \(D_i\) is the determinant obtained by replacing the \(i^{th}\) column of the coefficient matrix with the constant terms vector.

Step 2: Evaluate the statement

According to the formula of Cramer's Rule as presented in step 1, it can be seen that the numerator is \(D_i\) where \(D_i\) is the determinant obtained by replacing the \(i^{th}\) column of the coefficient matrix with the constant terms vector. The statement 'In Cramer's Rule, the numerator is the determinant of the coefficient matrix.' is therefore false as the numerator is not the determinant of the coefficient matrix but the determinant of a modified matrix where one column is replaced by the constant terms vector.

Key Concepts

Understanding Systems of Linear Equations

When we talk about 'systems of linear equations', we're referring to collections of equations that each represent a line. Together, they define a relationship between variables. For example, consider two equations with two variables, such as:
\[ x + y = 5 \] and
\[ 2x - 3y = -4. \]
These two equations together make up a system. Solving this system means finding the values of the variables (\( x \) and \( y \)) that satisfy both equations at the same time.
To solve such systems, various methods can be used, including substitution, elimination, and graphical representation. However, when we have larger systems or prefer an algebraic approach, methods like matrix operations or Cramer's Rule come in handy. With Cramer's Rule specifically, we can solve for each variable of the system when certain conditions are met—mainly, the system must have the same number of equations as unknowns, and the determinant of the coefficient matrix (more on this next) must not be zero.

The Determinant of a Matrix

The 'determinant of a matrix' is a special number that we can calculate from a square matrix. It is a value that gives us important information about the matrix, such as whether it has an inverse or what the volume of a parallelepiped defined by its rows (or columns) would be.
For example, a 2x2 matrix
\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
has a determinant calculated by
\[ \text{det}(A) = ad - bc. \]
The determinant of larger matrices is calculated using more complex methods, like expansion by minors or utilizing the Leibniz formula.
The determinant plays a pivotal role in Cramer's Rule, as it is used for both the denominator (\( D \)), which is the determinant of the coefficient matrix of the system, and the numerators (\( D_i \)), which are the determinants of matrices created by replacing a column of the coefficient matrix with the constant terms vector. The determinant tells us whether a unique solution exists—if the determinant is zero, the system does not have a unique solution.

Exploring the Coefficient Matrix

In the realm of linear algebra, the 'coefficient matrix' is a matrix consisting of the coefficients of the variables in a system of linear equations.
For example, in the system \[ \begin{align*} 3x + 4y &= 10, \ 2x - y &= 5, \end{align*} \]
the coefficient matrix is
\[ \begin{bmatrix} 3 & 4 \ 2 & -1 \end{bmatrix}. \]
This matrix holds a special place in solving systems because it represents the 'skeleton' of the system, stripping away the variables and leaving only the numbers that multiply them.
In the context of Cramer's Rule, the coefficient matrix is crucial as it forms the basis for calculating determinants that are essential in finding the variable values. The non-zero determinant of the coefficient matrix is a prerequisite for Cramer's Rule to be applicable.

Unpacking the Constant Terms Vector

The 'constant terms vector' is a column matrix that contains the constants from the right-hand side of each equation in a system of linear equations. For example, if we're dealing with a system like:
\[ \begin{align*} a_1x + b_1y &= c_1, \ a_2x + b_2y &= c_2, \end{align*} \]
the constant terms vector would be:
\[ \begin{bmatrix} c_1 \ c_2 \end{bmatrix}. \]
The constant terms vector is significant in Cramer's Rule because it replaces one of the columns in the coefficient matrix to calculate the determinants (\( D_i \)) used in the numerators for the solution of each variable. This vector is what relates the abstract coefficients to the actual quantities of the problem and thus drives towards the solution when plugged into the right place in the coefficient matrix.