Question
Find the determinant of the
diagonal. (Note that your answer will be a function of
diagonal. (Note that your answer will be a function of
Ask by Joseph Gibson. in the United States
Jan 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The determinant of matrix
Solution
The matrix
-
on the diagonal. -
s above the diagonal. -
s below the diagonal.
For any upper triangular matrix, the determinant is the product of the diagonal elements, regardless of the entries above the diagonal.
Given that all diagonal entries of
Final Answer:
After simplifying, the determinant is five to the nth power. Thus det A = 5ⁿ
After simplifying, the determinant is five to the nth power. Thus det A = 5ⁿ
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Simplify this solution The Deep Dive
To find the determinant of the matrix
Using properties of determinants, specifically for triangular matrices, the determinant is the product of the diagonal entries. However, our configuration has additional non-zero entries above the diagonal. We can utilize a recursive approach or apply a pattern recognition with determinants of smaller sizes.
Let’s denote
- For
, . - For
, the matrix becomes . - Thus,
.
- Thus,
For larger matrices, we could use the Laplace expansion or other properties, but we note that the determinant relates recursively to
With initial conditions
Ultimately, the solution to the recurrence yields:
Thus, we conclude:
