Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Question:**
5. Let
\[
A = \begin{pmatrix} 0 & k & 3 \\ 1 & -2 & 2 \\ 4 & 3 & 1 \end{pmatrix}
\]
Find the values of \( k \) where \( A \) is not invertible.
**Solution:**
To find the values of \( k \) that make the matrix \( A \) not invertible, we need to determine when the determinant of \( A \) is zero. A matrix is not invertible if its determinant is zero.
**Steps to Calculate the Determinant:**
1. The determinant of a 3x3 matrix
\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]
is given by:
\[
a(ei - fh) - b(di - fg) + c(dh - eg)
\]
2. Apply these steps specifically to the matrix \( A \):
- \( a = 0, b = k, c = 3 \)
- Second row: \( d = 1, e = -2, f = 2 \)
- Third row: \( g = 4, h = 3, i = 1 \)
3. Substituting these values into the formula:
\[
\text{det}(A) = 0((-2 \cdot 1) - (2 \cdot 3)) - k((1 \cdot 1) - (2 \cdot 4)) + 3((1 \cdot 3) - (-2 \cdot 4))
\]
4. Simplify each term:
- First term: \( 0 \times -8 = 0 \)
- Second term: \(-k(1 - 8) = -k(-7) = 7k\)
- Third term: \(3(3 + 8) = 3(11) = 33\)
5. Therefore, the determinant is:
\[
0 + 7k + 33 = 7k + 33
\]
6. Set the determinant equal to zero to find when the matrix is not invertible:
\[
7k + 33 = 0
\]
7](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c77c8ca-cf2a-49c8-86aa-e4a7d9346a8b%2Feeb3888d-cb13-4062-9285-cd67ee511fda%2Favp8l1c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
5. Let
\[
A = \begin{pmatrix} 0 & k & 3 \\ 1 & -2 & 2 \\ 4 & 3 & 1 \end{pmatrix}
\]
Find the values of \( k \) where \( A \) is not invertible.
**Solution:**
To find the values of \( k \) that make the matrix \( A \) not invertible, we need to determine when the determinant of \( A \) is zero. A matrix is not invertible if its determinant is zero.
**Steps to Calculate the Determinant:**
1. The determinant of a 3x3 matrix
\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]
is given by:
\[
a(ei - fh) - b(di - fg) + c(dh - eg)
\]
2. Apply these steps specifically to the matrix \( A \):
- \( a = 0, b = k, c = 3 \)
- Second row: \( d = 1, e = -2, f = 2 \)
- Third row: \( g = 4, h = 3, i = 1 \)
3. Substituting these values into the formula:
\[
\text{det}(A) = 0((-2 \cdot 1) - (2 \cdot 3)) - k((1 \cdot 1) - (2 \cdot 4)) + 3((1 \cdot 3) - (-2 \cdot 4))
\]
4. Simplify each term:
- First term: \( 0 \times -8 = 0 \)
- Second term: \(-k(1 - 8) = -k(-7) = 7k\)
- Third term: \(3(3 + 8) = 3(11) = 33\)
5. Therefore, the determinant is:
\[
0 + 7k + 33 = 7k + 33
\]
6. Set the determinant equal to zero to find when the matrix is not invertible:
\[
7k + 33 = 0
\]
7
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