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Chapter 6: Problem 60

In applying Cramer's Rule, what should you do if \(D=0 ?\)

Short Answer

Expert verified
If the determinant of the coefficient matrix, D, is 0 while applying Cramer's Rule, it means the system doesn't have a unique solution. Therefore Cramer's Rule can't be applied in such a scenario. The system itself could be either dependent or inconsistent.

Step by step solution

01

Understanding Cramer's Rule

Cramer's Rule is a method used to solve systems of linear equations by using the determinants of matrices. It involves two main steps: finding the determinant of the coefficients (D) and substituting the column vectors with the constants while calculating the determinants.
02

Consideration when D=0

If the determinant of the coefficient matrix (D) equals 0, it means that the coefficient matrix is singular, i.e., it does not have an inverse. This is because the determinant is a special number that can tell us a lot about a matrix, and one of the things it says is whether the matrix has an inverse. In such a case, Cramer's Rule cannot be applied.
03

Final Analysis

So, if D=0 during the use of Cramer's Rule, it should be concluded that the system of equations has no unique solution or an infinite number of solutions. Further, when D=0, the linear system could be either dependent or inconsistent and to determine which is the case a different method than Cramer's Rule should be utilized.

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Most popular questions from this chapter

Exercises \(85-87\) will help you prepare for the material covered in the next section. Use Gauss-Jordan elimination to solve the system: $$\left\\{\begin{array}{cc}-x-y-z=1 \\\4 x+5 y & =0 \\\y-3 z=0\end{array}\right.$$

What is meant by the order of a matrix? Give an example with your explanation.

The interesting and useful applications of matrix theory are nearly unlimited. Applications of matrices range from representing digital photographs to predicting long-range trends in the stock market. Members of the group should research an application of matrices that they find intriguing. The group should then present a seminar to the class about this application.

The figure shows the letter \(L\) in a rectangular coordinate system. (GRAPH CANNOT COPY) The figure can be represented by the matrix $$B=\left[\begin{array}{llllll}0 & 3 & 3 & 1 & 1 & 0 \\\0 & 0 & 1 & 1 & 5 & 5\end{array}\right]$$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The \(L\) is completed by connecting the last point in the matrix, \((0,5),\) to the starting point, \((0,0) .\) Use these ideas to solve Exercises \(53-60 .\) (In Exercises \(59-60\), be surre to refer to matrix \(B\) described in the second column on the previous page.) a. If \(A=\left[\begin{array}{ll}2 & 0 \\ 0 & 1\end{array}\right],\) find \(A B\) b. Graph the object represented by matrix \(A B\). What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(\vec{B}\) ?

Let $$\begin{aligned}&A=\left[\begin{array}{ll}1 & 0 \\\0 & 1\end{array}\right], \quad B=\left[\begin{array}{rr}1 & 0 \\\0 & -1\end{array}\right], \quad C=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]\\\&D=\left[\begin{array}{rr}-1 & 0 \\\0 & -1\end{array}\right]\end{aligned}$$ Find the product of the difference between \(A\) and \(B\) and the sum of \(C\) and \(D\)
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