Problem 38 - Precalculus: Functions and Graphs

Question

Use Cramer's rule, whenever applicable, to solve the system. $$\left\\{\begin{array}{r} 3 p-q=7 \\ -12 p+4 q=3 \end{array}\right.$$

Step-by-Step Solution

Answer

The system cannot be solved using Cramer's rule as the determinant is zero.

Step 1: Write the system in matrix form

To use Cramer's rule, first express the system of equations in matrix form: \[\begin{bmatrix}3 & -1 \-12 & 4\end{bmatrix}\begin{bmatrix}p \q\end{bmatrix} =\begin{bmatrix}7 \3\end{bmatrix}\]Where Matrix \(A = \begin{bmatrix} 3 & -1 \ -12 & 4 \end{bmatrix}\) and vector \(\mathbf{b} = \begin{bmatrix} 7 \ 3 \end{bmatrix}\).

Step 2: Compute the determinant of the coefficient matrix

Calculate the determinant of matrix \(A\), denoted as \(\det(A)\):\[\det(A) = \begin{vmatrix} 3 & -1 \ -12 & 4 \end{vmatrix} = (3)(4) - (-1)(-12) = 12 - 12 = 0\]Since the determinant is zero, Cramer's rule is not applicable because the system may not have a unique solution.

Key Concepts

Determinant

The determinant is a crucial value derived from a square matrix. It provides key insights into the matrix's properties. For a 2x2 matrix like \[\begin{bmatrix} a & b \c & d \end{bmatrix},\]the determinant is calculated using the formula: \[ ext{det}(A) = ad - bc.\]The determinant can tell us:

• If the matrix is invertible. A zero determinant means the matrix isn't invertible, or singular.
• About volume changes in linear transformations represented by the matrix.

In the context of solving a system of equations using Cramer's Rule, a non-zero determinant indicates a unique solution, whereas a zero determinant suggests the system might not have a unique solution, which is the case in this exercise.

Matrix Form

Matrix form is a method of organizing a system of linear equations for simplification and computation. It's particularly useful when employing techniques like Cramer's Rule.
In matrix form, a system of equationsdefines components as follows:

• The coefficient matrix (A), which contains the coefficients of the variables.
• The variable vector (usually denoted as \( \mathbf{x} \)), representing unknown variables.
• The constants vector (\( \mathbf{b} \)), containing the equations' constant terms.

For example, the system \[\begin{array}{r}3p - q = 7 \-12p + 4q = 3\end{array}\]can be expressed in matrix form as\[\begin{bmatrix}3 & -1 \-12 & 4\end{bmatrix} \begin{bmatrix}p \q\end{bmatrix} = \begin{bmatrix}7 \3\end{bmatrix}.\]This transforms the equations into a compact form that's easier to handle mathematically.

Unique Solution

In the context of systems of linear equations, a unique solution occurs when there is exactly one set of values for the variables that satisfies all equations in the system. Cramer's Rule specifically deals with systems of linear equations where the number of equations equals the number of variables.
A system has a unique solution if:

• The determinant of the coefficient matrix is non-zero. This implies the matrix is full-rank and invertible.
• All equations are consistent, meaning they work together without contradictions.

In the given exercise, the determinant was computed as zero, which typically indicates the absence of a unique solution. This means there could be either no solutions or an infinite number of solutions, and Cramer's Rule would not be applicable.