Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The original equation is verified after the calculations, hence it's correct.

Step 1: Calculate the first determinant

First, calculate the determinant of a matrix with elements \(w,cx,y,cz\). The formula for a 2x2 matrix determinant is \(\left|\begin{array}{ll}a & b \\c & d\end{array}\right|=ad-bc\). Substituting \(a=w, b=cx, c=y, d=cz\), compute the determinant to be \(w*cz-y*cx\).

Step 2: Calculate the second determinant

Next, calculate the determinant of the second matrix which has elements \(w,x,y,z\). Using the same formula \(\left|\begin{array}{ll}a & b \\c & d\end{array}\right|=ad-bc\), set \(a=w, b=x, c=y, d=z\) and compute the determinant to be \(w*z-y*x\).

Step 3: Multiply the second determinant by c

Now, multiply the second determinant by constant \(c\) to get \(c(wz-yx)\).

Step 4: Compare the results

After that, compare this product with the first determinant. Both are equal, hence the given property holds true. The original equation is correct.