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Chapter 8: Problem 47

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

Short Answer

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

Step by step solution

01

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).
02

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).
03

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.

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Most popular questions from this chapter

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 2 \end{array}\right]$$

Find the point of equilibrium of the demand and supply equations. The point of equilibrium is the price \(p\) and the number of units \(x\) that satisfy both the demand and supply equations. Demand \(\quad\) Supply \(p=140-0.00002 x \quad p=80+0.00001 x\)

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 2 x+3 y+5 z=4 \\ 3 x+5 y-9 z=7 \\ 5 x+9 y+17 z=13 \end{array}\right.$$

Solve for \(x\) $$\left|\begin{array}{cc} 2 x & -3 \\ -2 & 2 x \end{array}\right|=3$$

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} 4 u & -1 \\ -1 & 2 v \end{array}\right|$$
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