Chapter 11: Q. 50 (page 741)

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

Short Answer

Expert verified

The value of the determinant is $12$ .

Step by step solution

Step 1. Given Information

We are given,

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

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

Step 2. Solving the determinant

Using subtraction of determinant, we get

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

Since the two rows of second determinant is same, so

role="math" $|\begin{matrix} x + 3 & y + 6 & z + 9 \\ 3 u - 1 & 3 v - 2 & 3 w - 3 \\ 1 & 2 & 3 \end{matrix}| ⟹ |\begin{matrix} x + 3 & y + 6 & z + 9 \\ 3 u & 3 v & 3 w \\ 1 & 2 & 3 \end{matrix}| - 0$

Using the addition of determinant, we get

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

Step 3. Solving the determinant

Factoring out the GCF $3$ from the determinant, we get

role="math" localid="1647424003126" $= 3 |\begin{matrix} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{matrix}| + 3 |\begin{matrix} 1 & 2 & 3 \\ 3 u & 3 v & 3 w \\ 1 & 2 & 3 \end{matrix}| = 3 |\begin{matrix} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{matrix}| + 3 \times 0 = 3 \times 4 = 12$

Hence, the value of the determinant is $12$ .

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the determinant

Step 3. 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=4x+3y+6z+93u-13v-23w-3123

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solving the determinant

Step 3. Solving the determinant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Applied Mathematics

Logic and Functions

Calculus

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