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Chapter 8: Problem 46

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

Short Answer

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

Step by step solution

01

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 \(w*cz-y*cx\).
02

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 \(w*z-y*x\).
03

Multiply the second determinant by c

Now, multiply the second determinant by constant \(c\) to get \(c(wz-yx)\).
04

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.

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Most popular questions from this chapter

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{aligned} 6 b+15 a &=23.6 \\ 15 b+55 a &=48.8 \end{aligned}\right.$$

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