Compute the n x n determinants: X a a 8 a (a) a a a x ⠀⠀ : a a a ... ... 000 000 a X " (b) -t 0 1 -t 0 1 : : 0 00 0 ... ... ... ... ... ... 0 0 0 : 1 <-ao -a1 -a2 E -an-2 -an-1-t
Compute the n x n determinants: X a a 8 a (a) a a a x ⠀⠀ : a a a ... ... 000 000 a X " (b) -t 0 1 -t 0 1 : : 0 00 0 ... ... ... ... ... ... 0 0 0 : 1 <-ao -a1 -a2 E -an-2 -an-1-t
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Compute the \( n \times n \) determinants:**
**Matrix (a):**
\[
\begin{bmatrix}
x & a & a & \cdots & a \\
a & x & a & \cdots & a \\
a & a & x & \cdots & a \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a & a & a & \cdots & x \\
\end{bmatrix}
\]
**Matrix (b):**
\[
\begin{bmatrix}
-t & 0 & \cdots & 0 & -a_0 \\
1 & -t & \cdots & 0 & -a_1 \\
0 & 1 & \cdots & 0 & -a_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & -t & -a_{n-2} \\
0 & 0 & \cdots & 1 & -a_{n-1} - t \\
\end{bmatrix}
\]
**Explanation:**
1. **Matrix (a):** This is an \( n \times n \) matrix where each diagonal entry is \( x \), and all off-diagonal entries are \( a \). It has a symmetric structure.
2. **Matrix (b):** This is also an \( n \times n \) matrix showcasing a different pattern. The first column starts with \(-t\), with each subsequent entry increasing by 1 while managing a decrease of \( t \) in its diagonal entries. The last column contains terms of \(-a_0, -a_1, \ldots\), and the bottom right entry is written as \(-a_{n-1} - t\).
Both matrices are part of a problem in computing their determinants, highlighting symmetrical and tridiagonal-like patterns.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F7c5559cb-1cfc-41ae-aad6-07fb84fc55c6%2Fcxq49t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Compute the \( n \times n \) determinants:**
**Matrix (a):**
\[
\begin{bmatrix}
x & a & a & \cdots & a \\
a & x & a & \cdots & a \\
a & a & x & \cdots & a \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a & a & a & \cdots & x \\
\end{bmatrix}
\]
**Matrix (b):**
\[
\begin{bmatrix}
-t & 0 & \cdots & 0 & -a_0 \\
1 & -t & \cdots & 0 & -a_1 \\
0 & 1 & \cdots & 0 & -a_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & -t & -a_{n-2} \\
0 & 0 & \cdots & 1 & -a_{n-1} - t \\
\end{bmatrix}
\]
**Explanation:**
1. **Matrix (a):** This is an \( n \times n \) matrix where each diagonal entry is \( x \), and all off-diagonal entries are \( a \). It has a symmetric structure.
2. **Matrix (b):** This is also an \( n \times n \) matrix showcasing a different pattern. The first column starts with \(-t\), with each subsequent entry increasing by 1 while managing a decrease of \( t \) in its diagonal entries. The last column contains terms of \(-a_0, -a_1, \ldots\), and the bottom right entry is written as \(-a_{n-1} - t\).
Both matrices are part of a problem in computing their determinants, highlighting symmetrical and tridiagonal-like patterns.
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