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Chapter 3: Problem 14

Use Cramer's Rule to solve (if possible) the system of linear equations. $$\begin{aligned} &13 x-6 y=17\\\ &26 x-12 y=8 \end{aligned}$$

Short Answer

Expert verified
The system of equations does not have a unique solution, thus Cramer's Rule cannot be applied.

Step by step solution

01

Identify the Coefficient Matrix and the Constant Vector

Write out the system of equations in matrix form. The coefficient matrix \(A\) is \(\begin{bmatrix}13 & -6\\ 26 & -12\end{bmatrix}\) and the constant vector \(b\) is \(\begin{bmatrix}17\\ 8\end{bmatrix}\).
02

Calculate the Determinant of the Coefficient Matrix

The determinant of a 2x2 matrix \(\begin{bmatrix}a & b\\ c & d\end{bmatrix}\) is given by \(ad-bc\). So, the determinant of A (\(|A|\)) is \(13*(-12) - (-6)*26 = -156 - (-156) = 0\).
03

Verify the Conditions for Cramer's Rule

For Cramer's Rule to be used, the determinant of the coefficient matrix should not be zero. However, in this case, \(|A|=0\). So, Cramer's Rule cannot be applied and the system has no unique solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Linear Equations
A system of linear equations is a set of two or more equations each with a common set of variables. In our example, we have two equations with variables \( x \) and \( y \):
  • 13x - 6y = 17
  • 26x - 12y = 8
This system can represent real-world problems where relationships between quantities need to be solved. When represented graphically, each equation forms a line, and the solution, if it exists, corresponds to the intersection points of these lines.
The solutions can be:
  • Unique (where lines intersect at exactly one point)
  • Infinite (where lines overlap)
  • No solution (where lines are parallel)
Understanding the nature of solutions helps us direct our solving strategy.
Determinant
The determinant is a special number that can be calculated from a square matrix. It provides crucial information about the matrix, such as whether it's invertible.
For a 2x2 matrix, the determinant is calculated using: \[|A| = ad - bc\]For our coefficient matrix \(A = \begin{bmatrix}13 & -6 \ 26 & -12\end{bmatrix}\), the determinant is zero:
  • \(13 \times (-12) - (-6) \times 26 = -156 + 156 = 0\)
This zero determinant indicates that the matrix \(A\) doesn't have an inverse and the system of equations doesn't have a unique solution.
Coefficient Matrix
The coefficient matrix is a square matrix made by aligning the coefficients of the variables in a system of linear equations. In our example, the coefficient matrix is:
\[A = \begin{bmatrix}13 & -6 \ 26 & -12\end{bmatrix}\]Each element represents a coefficient from the system of equations. This matrix plays a critical role in various solving techniques like matrix inversion and Cramer's Rule. If its determinant is zero, as in our case, it tells us that the system either has no solution or infinitely many solutions.
Recognizing the coefficient matrix helps in evaluating and applying the correct methods for solving systems of equations.

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Most popular questions from this chapter

True or False? Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text. (a) If \(A\) is an \(n \times n\) matrix and \(c\) is a nonzero scalar, then the determinant of the matrix \(c A\) is \(n c \cdot \operatorname{det}(A)\) (b) If \(A\) is an invertible matrix, then the determinant of \(A^{-1}\) is equal to the reciprocal of the determinant of \(A\) (c) If \(A\) is an invertible \(n \times n\) matrix, then \(A x=b\) has a unique solution for every b.

Guided Proof Prove Property 2 of Theorem 3.3 : When \(B\) is obtained from \(A\) by adding a multiple of a row of \(A\) to another row of \(A, \operatorname{det (B)=\operatorname{det}(A)\) Getting Started: To prove that the determinant of \(B\) is equal to the determinant of \(A,\) you need to show that their respective cofactor expansions are equal. (i) Begin by letting \(B\) be the matrix obtained by adding \(c\) times the \(j\) th row of \(A\) to the \(i\) th row of \(A\) (ii) Find the determinant of \(B\) by expanding in this ith row. (iii) Distribute and then group the terms containing a coefficient of \(c\) and those not containing a coefficient of \(c\) (iv) Show that the sum of the terms not containing a coefficient of \(c\) is the determinant of \(A,\) and the sum of the terms containing a coefficient of \(c\) is equal to \(0 .\)

Determine whether the points are coplanar. $$(-3,-2,-1),(2,-1,-2),(-3,-1,-2),(3,2,1)$$

Determine whether the matrix is orthogonal. An invertible square matrix \(A\) is orthogonal when \(A^{-1}=A^{T}\). $$\left[\begin{array}{rrr}1 / \sqrt{2} & 0 & -1 / \sqrt{2} \\\0 & 1 & 0 \\\1 / \sqrt{2} & 0 & 1 / \sqrt{2}\end{array}\right]$$

Find an equation of the plane passing through the points. $$(3,2,-2),(3,-2,2),(-3,-2,-2)$$
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