Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Calculate the determinant

The determinant of a 3x3 matrix can be calculated using the formula: \(Det = a(ei−fh)−b(di−fg)+c(dh−eg)\). 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:\((y-x)(z-x)(z-y)\).

Step 3: Verification

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