Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

Upon calculating the determinants and comparing them, we do verify that the determinant of the first matrix equals the negative of the determinant of the second.

Step 1: Calculation of the first determinant

We calculate the determinant of the first matrix (\( w, x, y, z \)). The determinant, often represented as Det or | |, of a 2x2 matrix can be calculated as (ad - bc), where a, b, c and d are elements of the matrix. In this case, we find (w*z - x*y).

Step 2: Calculation of the second determinant

Next, we calculate the determinant of the second matrix (\( y, z, w, x \)). Using the same formula above, we find (y*x - z*w).

Step 3: Comparison of the determinants

The exercise, initially, postulates that the first determinant is the negative of the second one. So now, it is required to verify if (w*z - x*y) = -(y*x - z*w). Upon simplifying the right side of the equation, we notice that it indeed equals w*z - x*y, thereby confirming that the initial postulate regrading the determinants of the matrices holds true.