Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} 3 x^{2} & -3 y^{2} \\ 1 & 1 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the given matrix is \(3x^2 + 3y^2\).

Step 1: Identify the elements of the matrix

Given the matrix \[\left|\begin{array}{cc}3 x^{2} & -3 y^{2} \1 & 1\end{array}\right|\], the elements are: a = \(3x^2\), b = \(-3y^2\), c = 1, d = 1.

Step 2: Apply the formula of determinant of 2x2 matrix

By applying the formula mentioned in the analysis, determinant is equal to \(ad - bc\). Substitute the values in the formula and we get: Determinant = \(3x^2*1 - 1*(-3y^2) = 3x^2 + 3y^2\).

Step 3: Simplify the solution

You don’t need to simplify the result in this case because the result is already in the simplest form. Therefore, the determinant of the given matrix is \(3x^2 + 3y^2\).