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Chapter 3: Problem 71

Evaluate the determinants to verify the equation. $$\left|\begin{array}{lll} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}\right|=(y-x)(z-x)(z-y)$$

Short Answer

Expert verified
After expanding and simplifying the determinant of the given 3x3 matrix, it is verified that |A| equals to (y-x)(z-x)(z-y).

Step by step solution

01

Identify the Matrix

Given matrix A is \[ \left|\begin{array}{ccc}1 & x & x^{2} \1 & y & y^{2} \1 & z & z^{2} \end{array}\right| \]
02

Apply the determinant formula

To calculate the determinant (denoted as |A|), let's expand along the first column. The determinant formula becomes: \[ |A| = 1 \cdot \left|\begin{array}{ll}y & y^{2} \z & z^{2} \end{array}\right| - 1 \cdot \left|\begin{array}{ll}x & x^{2} \z & z^{2} \end{array}\right| + 1 \cdot \left|\begin{array}{ll}x & x^{2} \y & y^{2} \end{array}\right|. \]
03

Calculate the 2x2 determinants

In order to calculate the determinant of original matrix (3x3), we need to calculate the determinant for these submatrices (each 2x2) first. The calculation is as follows: \[|A| = 1(y*z^{2} - y^{2}*z) - 1(x*z^{2} - x^{2}*z) + 1(x*y^{2} - x^{2}*y) \] Simplify this to get \( |A| = z*y^{2} - y*z^{2} - z*x^{2} + x*z^{2} + x*y^{2} - x^{2}*y \)
04

Reorder and Factor

Rearrange and factor out common factors from the expression above: \( |A| = y^{2}(z - x) - z^{2}(y - x) + x^{2}(y - z) \) This further simplifies to \( |A| = (y-x)(z-x)(z-y) \) which verifies our given equation.

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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. $$\begin{array}{l} 4 x_{1}-x_{2}+x_{3}=-5 \\ 2 x_{1}+2 x_{2}+3 x_{3}=10 \\ 5 x_{1}-2 x_{2}+6 x_{3}=1 \end{array}$$

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

Determine whether the points are coplanar $$(0,0,-1),(0,-1,0),(1,1,0),(2,1,2)$$

Prove that if \(A\) is an orthogonal matrix, then \(|A|=\pm 1\)

Use a graphing utility or computer software program with matrix capabilities to find the eigenvalues of the matrix. Then find the corresponding eigenvectors. $$\left[\begin{array}{rr} 4 & 3 \\ -3 & -2 \end{array}\right]$$
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