Q. 62 Interchange columns 1聽and 3聽of... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 62 (page 741)

Interchange columns $1$ and $3$ of a $3$ by $3$ determinant. Show that the value of the new determinant is $- 1$ times the value of the original determinant.

Short Answer

Expert verified

It is proved thatby interchanging the columns the value of the new determinant is $- 1$ times the value of the original determinant.

Step by step solution

Step 1. Given Information

We are given a determinant and we have to show that by interchanging the columns the value of the new determinant is $- 1$ times the value of the original determinant.

Step 2. Proving the statement

Let the determinant be,

$|\begin{array}{l} a & b & c \\ d & e & f \\ g & h & i \end{array}|$

Expanding the determinant, we get

$|\begin{array}{l} a & b & c \\ d & e & f \\ g & h & i \end{array}| = (- 1)^{1 + 1} 脳 a 脳 |\begin{matrix} e & f \\ h & i \end{matrix}| + (- 1)^{1 + 2} 脳 b 脳 |\begin{array}{l} d & f \\ g & i \end{array}| + (- 1)^{1 + 2} 脳 c 脳 |\begin{array}{l} d & e \\ g & h \end{array}| = a e i - a f h - b d i + b f g + c d h - c e g$

Now, Interchanging the columns $1$ and $3$ , we get

$|\begin{matrix} c & b & a \\ f & e & d \\ i & h & g \end{matrix}|$

Step 3. Evaluating the determinant

Evaluating the changed determinant, we get

$|\begin{matrix} c & b & a \\ f & e & d \\ i & h & g \end{matrix}| = (- 1)^{1 + 1} 脳 c 脳 |\begin{array}{l} e & d \\ h & g \end{array}| + (- 1)^{1 + 2} 脳 b 脳 |\begin{array}{l} f & d \\ i & g \end{array}| + (- 1)^{1 + 3} 脳 a 脳 |\begin{array}{l} f & e \\ i & h \end{array}| = g e c - h d c - g f b + i b d + h f a - a e i = - (a e i - a f h - b d i + b f g + c d h - c e g) = - |\begin{array}{l} a & b & c \\ d & e & f \\ g & h & i \end{array}|$

Hence Proved.

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91影视

Interchange columns $1$ and $3$ of a $3$ by $3$ determinant. Show that the value of the new determinant is $- 1$ times the value of the original determinant.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

Step 3. Evaluating the determinant

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Interchange columns 1and 3of a 3by 3determinant. Show that the value of the new determinant is -1times the value of the original determinant.

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Step by step solution

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Interchange columns $1$ and $3$ of a $3$ by $3$ determinant. Show that the value of the new determinant is $- 1$ times the value of the original determinant.