Chapter 11: Q. 47 (page 741)

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that
$|\begin{array}{llr} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}| = 4 |\begin{matrix} 1 & 2 & 3 \\ x - 3 & y - 6 & z - 9 \\ 2 u & 2 v & 2 w \end{matrix}|$

Short Answer

Expert verified

The value of the determinant is $8$ .

Step by step solution

Step 1. Given Information

We are given,

$|\begin{array}{l} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}| = 4 |\begin{matrix} 1 & 2 & 3 \\ x - 3 & y - 6 & z - 9 \\ 2 u & 2 v & 2 w \end{matrix}|$

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

Step 2. Solving the determinant

Factoring out the GCF $2$ , we get

localid="1647357556454" $|\begin{matrix} 1 & 2 & 3 \\ x - 3 & y - 6 & z - 9 \\ 2 u & 2 v & 2 w \end{matrix}| = 2 |\begin{matrix} 1 & 2 & 3 \\ x - 3 & y - 6 & z - 9 \\ u & v & w \end{matrix}|$

Performing the row operation localid="1647357674207" $R_{2} = R_{2} + 3 R_{1}$ ,

$2 |\begin{matrix} 1 & 2 & 3 \\ x - 3 & y - 6 & z - 9 \\ u & v & w \end{matrix}| = 2 |\begin{matrix} 1 & 2 & 3 \\ x & y & z \\ u & v & w \end{matrix}|$

Interchanging the rows localid="1647417532460" $R_{1} and R_{2}$ , we get

localid="1647417583543" $= - 2 |\begin{array}{l} x & y & z \\ 1 & 2 & 3 \\ u & v & w \end{array}|$

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

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

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In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that
$|\begin{array}{llr} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}| = 4 |\begin{matrix} 1 & 2 & 3 \\ x - 3 & y - 6 & z - 9 \\ 2 u & 2 v & 2 w \end{matrix}|$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the determinant

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In Problems 43–50, use properties of determinants to find the value of each determinant if it is known thatxyzuvw123=4123x-3y-6z-92u2v2w

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the determinant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Calculus

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 43–50, use properties of determinants to find the value of each determinant if it is known that
$|\begin{array}{llr} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}| = 4 |\begin{matrix} 1 & 2 & 3 \\ x - 3 & y - 6 & z - 9 \\ 2 u & 2 v & 2 w \end{matrix}|$