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Chapter 3: Problem 62

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} e^{-x} & x e^{-x} \\ -e^{-x} & (1-x) e^{-x} \end{array}\right|$$

Short Answer

Expert verified
The determinant of the given matrix is \(xe^{-2x}\).

Step by step solution

01

Identify Elements of the Matrix

The matrix we are given is \(\left|\begin{array}{cc}a & b \ c & d \end{array}\right|\). In this case, \(a=e^{-x}\), \(b=x e^{-x}\), \(c=-e^{-x}\), and \(d=(1-x) e^{-x}\).
02

Calculate the Determinant

The formula for calculating the determinant of a 2x2 matrix is \(ad - bc\). Therefore, substitute the identified elements from Step 1 into this formula, which will result in \(e^{-x}*(1-x)e^{-x}-(-e^{-x}*x*e^{-x})\).
03

Simplify the Expression

Simplify the expression resulting from Step 2 to find the determinant. It will become \(e^{-2x} - e^{-2x} + xe^{-2x}\), which when simplified further is equal to \(xe^{-2x}\).

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Most popular questions from this chapter

Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{aligned} -x_{1}-x_{2} \quad+x_{4}=-8 \\ 3 x_{1}+5 x_{2}+5 x_{3} \quad=24 \\ 2 x_{3}+x_{4} =-6 \\ -2 x_{1}-3 x_{2}-3 x_{3} \quad=-15 \end{aligned}$$

Use Cramer's Rule to solve the system of linear equations, if possible. $$\begin{array}{l} 18 x_{1}+12 x_{2}=13 \\ 30 x_{1}+24 x_{2}=23 \end{array}$$

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 the statement is not true in all cases or cite an appropriate statement from the text (a) The characteristic equation of the matrix \(A=\left[\begin{array}{rr}2 & -1 \\\ 1 & 0\end{array}\right]\) yields eigenvalues \(\lambda_{1}=\lambda_{2}=1\) (b) The matrix \(A=\left[\begin{array}{rr}4 & -2 \\ -1 & 0\end{array}\right]\) has irrational eigenvalues \(\lambda_{1}=2+\sqrt{6}\) and \(\lambda_{2}=2-\sqrt{6}\)

Verify that \(\lambda_{i}\) is an eigenvalue of \(A\) and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. $$\begin{array}{l} A=\left[\begin{array}{ll} 4 & 3 \\ 1 & 2 \end{array}\right] ; \quad \lambda_{1}=5, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 3 \\ 1 \end{array}\right] \\ \lambda_{2}=1, \quad \mathbf{x}_{2}=\left[\begin{array}{r} -1 \\ 1 \end{array}\right] \end{array}$$

Find an equation of the plane passing through the three points. $$(1,2,7),(4,4,2),(3,3,4)$$
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