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Chapter 8: Problem 45

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

Short Answer

Expert verified
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 by step solution

01

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).
02

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).
03

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.

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