Sanders Fitzgerald
06/23/2024 · Elementary School
that \( \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|=x y \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To evaluate the determinant
\[
D = \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|,
\]
we can use the properties of determinants. We will perform row operations to simplify the determinant.
First, we can subtract the first row from the second and third rows:
\[
D = \left|\begin{array}{ccc}1 & 1 & 1 \\ 0 & x & 0 \\ 0 & 0 & y\end{array}\right|.
\]
Now, we can compute the determinant of this upper triangular matrix. The determinant of an upper triangular matrix is the product of its diagonal entries:
\[
D = 1 \cdot x \cdot y = xy.
\]
Thus, we have shown that
\[
\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right| = xy.
\]
Therefore, the final result is
\[
\boxed{xy}.
\]
Quick Answer
The determinant equals \( xy \).
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit