Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The determinants of given matrices are equal, thus verifying the equation.

Step 1: Calculate Determinant of First Matrix

To calculate the determinant of the first matrix, subtract the product of the elements of the second diagonal (x*y) from the product of the elements of the main diagonal (w*z). That is, by applying (a*d - b*c), we get (w*z - x*y).

Step 2: Calculate Determinant of Second Matrix

To calculate the determinant of the second matrix, subtract the product of the second diagonal elements ((x+cw)*y) from the product of the main diagonal elements (w*(z+cy)). Here, the multiplication rule is used, w*z + w*cy - x*y - cw*y. After expanding and simplifying, we get (w*z - x*y).

Step 3: Comparing the Determinants

We compare the determinants of the first and second matrices. It is clear from the above calculations that the determinant of the first matrix is equal to the determinant of the second matrix.