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Chapter 8: Problem 37

Determine whether the statement is true or false. Justify your answer. Cramer's Rule cannot be used to solve a system of linear equations when the determinant of the coefficient matrix is zero.

Short Answer

Expert verified
The statement is true. Cramer's Rule cannot be used to solve a system of linear equations when the determinant of the coefficient matrix is zero because the system may not have a unique solution in this case.

Step by step solution

01

Analyzing the Statement

Cramer's Rule can be used to solve a system of linear equations by using determinants. According to Cramer's Rule, the system of linear equations has a unique solution if and only if the determinant of the coefficient matrix of the system of equations is not zero.
02

Verifying the Statement

The statement 'Cramer's Rule cannot be used to solve a system of linear equations when the determinant of the coefficient matrix is zero' is evaluated. It aligns with the condition of Cramer's Rule. When the determinant of the coefficient matrix is zero, the system of equations may either have infinite solutions or no solutions, hence Cramer's Rule cannot be used.
03

Statement Evaluation

Given this information, it can be concluded that the statement is true. If the determinant of the coefficient matrix of a system is zero, Cramer's Rule cannot be used to find a unique solution for that system.

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Most popular questions from this chapter

Solve for \(x\) $$\left|\begin{array}{ll} x & 2 \\ 1 & x \end{array}\right|=2$$

Determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, then the determinant will always be zero.

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{ll} 5 & -1 \\ 2 & -1 \end{array}\right]$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{rr} 7 x-3 y & +2 w=41 \\ -2 x+y & -w=-13 \\ 4 x+z-2 w & =12 \\ -x+y-x & =-8 \end{array}\right.$$

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{2 x-7}{3 x+2}$$
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