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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 49 (page 741)

In Problems 43鈥�50, use properties of determinants to find the value of each determinant if it is known that
$|\begin{array}{llr} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}| = 4 |\begin{matrix} 1 & 2 & 3 \\ 2 x & 2 y & 2 z \\ u - 1 & v - 2 & w - 3 \end{matrix}|$

Short Answer

Expert verified

The value of the determinant is $8$ .

Step by step solution

01

Step 1. Given Information

We are given,

$|\begin{array}{llr} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}| = 4 |\begin{matrix} 1 & 2 & 3 \\ 2 x & 2 y & 2 z \\ u - 1 & v - 2 & w - 3 \end{matrix}|$

We have to solve the given determinant using properties of determinants.

02

Step 2. Solving the determinant

Factoring out the GCF 2 $2$ , we get

$|\begin{matrix} 1 & 2 & 3 \\ 2 x & 2 y & 2 z \\ u - 1 & v - 2 & w - 3 \end{matrix}| = 2 |\begin{matrix} 1 & 2 & 3 \\ x & y & z \\ u - 1 & v - 2 & w - 3 \end{matrix}|$

Performing row operation $R_{3} = R_{3} + R_{1}$ , we get

$= 2 |\begin{matrix} 1 & 2 & 3 \\ x & y & z \\ u & v & w \end{matrix}|$

Interchanging the rows $R_{1} and R_{2}$ , we get

role="math" localid="1647417553137" $= - 2 |\begin{array}{l} x & y & z \\ 1 & 2 & 3 \\ u & v & w \end{array}|$

Interchanging the rows $R_{2} and R_{3}$ , we get

$= 2 |\begin{array}{l} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}| = 2 脳 4 = 8$

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Most popular questions from this chapter

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} \begin{array}{rcl} 3 x + 3 y + 2 z & = & 4 \\ x - 3 y + z & = & 10 \end{array} \\ 5 x - 2 y - 3 z = 8 \end{matrix} x = 2, y = - 2, z = 2; (2, - 2, 2)$

The perimeter of a rectangle is $16$ inches, and its area is $15 i n c h e s^{2}$ . What are its dimensions?

solve each system of equations. If the system has no solution, say that it is inconsistent.

$2 x + y = 4 - 2 y + 4 z = 0 3 x - 2 z = - 11$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$2 x - 3 y - z = 0 3 x + 2 y + 2 z = 2 x + 5 y + 3 z = 2$

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$y = 3 x - 5; x^{2} + y^{2} = 5$

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