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Chapter 6: Problem 56

Describe the determinants \(D_{x}\) and \(D_{y}\) in terms of the coefficients and constants in a system of two equations in two variables.

Short Answer

Expert verified
The determinants \(D_{x}\) and \(D_{y}\) in a system of two equations are given by \(D_{x} = ed - bf\) and \(D_{y} = af - ec\), where \(ax + by = e\) and \(cx + dy = f\) are the two equations.

Step by step solution

01

Understanding the notations

Assume we have a system of two linear equations in the form:\[ ax + by = e \]\[ cx + dy = f \]The determinant of the system \(D\) is computed from the coefficients of x and y as:\[ D = \left|\begin{array}{ll} a & b \ c & d \end{array} \right| = ad - bc \]
02

Determinant \(D_{x}\)

The determinant \(D_{x}\) is computed by replacing the coefficients of x in the determinant of the system with the constants on the right side of the equations:\[ D_{x} = \left|\begin{array}{ll} e & b \ f & d \end{array} \right| = ed - bf \]
03

Determinant \(D_{y}\)

Similarly, the determinant \(D_{y}\) is computed by replacing the coefficients of y in the determinant of the system with the constants on the right side of the equations:\[ D_{y} = \left|\begin{array}{ll} a & e \ c & f \end{array} \right| = af - ec \]

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Most popular questions from this chapter

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows If an operation is not defined, state the reason. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$A(B+C)$$

Explain how to evaluate a third-order determinant.

The process of solving a linear system in three variables using Cramer's Rule can involve tedious computation. Is there a way of speeding up this process, perhaps using Cramer's Rule to find the value for only one of the variables? Describe how this process might work, presenting a specific example with your description. Remember that your goal is still to find the value for each variable in the system.

Find \(A^{-1}\) and check. $$A=\left[\begin{array}{cc}e^{2 x} & -e^{x} \\\e^{3 x} & e^{2 x}\end{array}\right]$$

Explain why a matrix that does not have the same number of rows and columns cannot have a multiplicative inverse.
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