College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.5: Determinants
Problem 79E
Related questions
Question
![### Finding Values of \( a \) for Determinant of Matrix \( A \)
In this exercise, we are given a matrix \(A\) and tasked with finding the values of the variable \( a \) that make the determinant of \( A \) equal to zero. The matrix \( A \) is defined as:
\[ A = \begin{bmatrix}
a & 5 & 7 \\
a & -2 & 4 \\
4 & 0 & a \\
\end{bmatrix} \]
#### Problem Statement
Find all values of \( a \) that make \(|A| = 0\). Give your answer as a comma-separated list.
#### Input Box:
Values of \( a \): [ __________ ]
### Instructions
To solve this problem, you need to calculate the determinant of the given 3x3 matrix \( A \). The determinant of a 3x3 matrix:
\[ \begin{vmatrix}
a & 5 & 7 \\
a & -2 & 4 \\
4 & 0 & a \\
\end{vmatrix} \]
is found using the formula:
\[
\text{det}(A) = a \cdot \text{det} \begin{bmatrix}
-2 & 4 \\
0 & a
\end{bmatrix}
- 5 \cdot \text{det} \begin{bmatrix}
a & 4 \\
4 & a
\end{bmatrix}
+ 7 \cdot \text{det} \begin{bmatrix}
a & -2 \\
4 & 0
\end{bmatrix}
\]
By evaluating the above, we can find the values of \( a \) that make the determinant zero. Enter the resulting values of \( a \) in the input box provided as a comma-separated list.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e366a85-c23b-450a-9bc7-d2ea0be52d3f%2F5b74d13b-8b29-4137-b6aa-26f79c3d1f0a%2Fw23ui2e_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Values of \( a \) for Determinant of Matrix \( A \)
In this exercise, we are given a matrix \(A\) and tasked with finding the values of the variable \( a \) that make the determinant of \( A \) equal to zero. The matrix \( A \) is defined as:
\[ A = \begin{bmatrix}
a & 5 & 7 \\
a & -2 & 4 \\
4 & 0 & a \\
\end{bmatrix} \]
#### Problem Statement
Find all values of \( a \) that make \(|A| = 0\). Give your answer as a comma-separated list.
#### Input Box:
Values of \( a \): [ __________ ]
### Instructions
To solve this problem, you need to calculate the determinant of the given 3x3 matrix \( A \). The determinant of a 3x3 matrix:
\[ \begin{vmatrix}
a & 5 & 7 \\
a & -2 & 4 \\
4 & 0 & a \\
\end{vmatrix} \]
is found using the formula:
\[
\text{det}(A) = a \cdot \text{det} \begin{bmatrix}
-2 & 4 \\
0 & a
\end{bmatrix}
- 5 \cdot \text{det} \begin{bmatrix}
a & 4 \\
4 & a
\end{bmatrix}
+ 7 \cdot \text{det} \begin{bmatrix}
a & -2 \\
4 & 0
\end{bmatrix}
\]
By evaluating the above, we can find the values of \( a \) that make the determinant zero. Enter the resulting values of \( a \) in the input box provided as a comma-separated list.
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