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Chapter 3: Problem 67

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=-\left|\begin{array}{ll} y & z \\ w & x \end{array}\right|$$

Short Answer

Expert verified
The given equation can be verified by calculating the determinants of both matrices and proving that the determinant of the first matrix is indeed the negation of the determinant of the second one.

Step by step solution

01

Calculate the determinant of the first matrix

The determinant of the first matrix, is calculated as (w*z) - (x*y). Let's denote this as \(D_1\).
02

Calculate the determinant of the second matrix

Similarly, the determinant of the second matrix is performed as (y*x) - (z*w). Let's denote this as \(D_2\).
03

Compare the results

According to the provided equation, \(D_1\) should be equal to the negation of \(D_2\). If the equation stands true, then our calculations are correct.

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Most popular questions from this chapter

Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{ll} 4 & -5 \\ 2 & -3 \end{array}\right]$$

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

Use a graphing utility or computer software program with matrix capabilities to find the eigenvalues of the matrix. Then find the corresponding eigenvectors. $$\left[\begin{array}{rrr} 4 & -2 & -2 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right]$$

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. $$\begin{array}{l} 2 x_{1}-x_{2}=-10 \\ 3 x_{1}+2 x_{2}=-1 \end{array}$$
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