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Chapter 8: Problem 49

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
The determinant of the matrix verifies the equation as \((y-x)(z-x)(z-y)\).

Step by step solution

01

Understanding the matrix

The matrix contains three rows, with each column containing the variables x, y, and z raised to 0, 1, and 2 respectively.
02

Calculate the determinant

The determinant of a 3x3 matrix can be calculated using the formula: \(Det = a(ei鈭抐h)鈭抌(di鈭抐g)+c(dh鈭抏g)\). Replacing these with the elements of the given matrix, we get:\( Det = 1[(y)*z^{2}-z*y^{2}] -x [1*z^{2}-z*1] + x^{2} [1*y-y] \). With simplification this gives:\(1[z^{2}y-y^{2}z]-x[z^{2}-z]+ x^{2}[y-y]\). The term with the square falls out because it is 0. This simplifies to:\(z^{2}y-y^{2}z-z^{2}x+z^{2}+zx-yx\). Reorganizing gives us\((y-x)z^{2}+(x-z)yz+(z-y)yx\). Factor out terms to get the final answer:\((y-x)(z-x)(z-y)\).
03

Verification

The calculated determinant is exactly the same as the given equation, hence it verifies the equation.

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Most popular questions from this chapter

Solve for \(x\) $$\left|\begin{array}{rrr} 1 & 2 & x \\ -1 & 3 & 2 \\ 3 & -2 & 1 \end{array}\right|=0$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 2 x+3 y+5 z=4 \\ 3 x+5 y-9 z=7 \\ 5 x+9 y+17 z=13 \end{array}\right.$$

Determine whether the statement is true or false. Justify your answer. Writing Briefly explain whether or not it is possible for a consistent system of linear equations to have exactly two solutions.

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

Determine whether the statement is true or false. Justify your answer. If a system of linear equations has two distinct solutions, then it has an infinite number of solutions.
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