Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Task**
Evaluate the determinant of the following 3x3 matrix:
\[
\begin{vmatrix}
4 & 8 & 1 \\
7 & -6 & 0 \\
3 & 2 & 1
\end{vmatrix}
\]
**Solution**
The determinant of a 3x3 matrix \(\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}\) is calculated using the formula:
\[
\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\]
For the given matrix:
- \( a = 4, b = 8, c = 1 \)
- \( d = 7, e = -6, f = 0 \)
- \( g = 3, h = 2, i = 1 \)
Substitute these values into the formula:
\[
\text{det} = 4((-6)(1) - (0)(2)) - 8((7)(1) - (0)(3)) + 1((7)(2) - (-6)(3))
\]
\[
= 4(-6) - 8(7) + 1(14 + 18)
\]
\[
= -24 - 56 + 32
\]
\[
= -80 + 32
\]
\[
= -48
\]
The result is \(-48\), however, the image shows a calculation with the result \(-1\), indicating an error was made in that example.
**Explanation of Error**
The given attempt to solve resulted in \(-1\), which is incorrect based on the described formula and steps. The correct determinant of the matrix should be \(-48\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F370ea171-8bb9-4b07-85c5-04ae9f865ef9%2Fb0406e86-a75f-4e9f-b887-8ca98246b484%2Fxbylr8j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task**
Evaluate the determinant of the following 3x3 matrix:
\[
\begin{vmatrix}
4 & 8 & 1 \\
7 & -6 & 0 \\
3 & 2 & 1
\end{vmatrix}
\]
**Solution**
The determinant of a 3x3 matrix \(\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}\) is calculated using the formula:
\[
\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\]
For the given matrix:
- \( a = 4, b = 8, c = 1 \)
- \( d = 7, e = -6, f = 0 \)
- \( g = 3, h = 2, i = 1 \)
Substitute these values into the formula:
\[
\text{det} = 4((-6)(1) - (0)(2)) - 8((7)(1) - (0)(3)) + 1((7)(2) - (-6)(3))
\]
\[
= 4(-6) - 8(7) + 1(14 + 18)
\]
\[
= -24 - 56 + 32
\]
\[
= -80 + 32
\]
\[
= -48
\]
The result is \(-48\), however, the image shows a calculation with the result \(-1\), indicating an error was made in that example.
**Explanation of Error**
The given attempt to solve resulted in \(-1\), which is incorrect based on the described formula and steps. The correct determinant of the matrix should be \(-48\).
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