In each of dhe tollowing , tactor the matrix A product xDX", where D as diagoral. in to a (a) 2 -8 A -4 2 2 (b) A = - | (C) A = -2 1 3 - 1 O O
In each of dhe tollowing , tactor the matrix A product xDX", where D as diagoral. in to a (a) 2 -8 A -4 2 2 (b) A = - | (C) A = -2 1 3 - 1 O O
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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Question
![**Diagonalization of Matrices**
In each of the following problems, factor the matrix \( A \) into a product \( XDX^{-1} \), where \( D \) is a diagonal matrix.
(a)
\[
A = \begin{bmatrix}
2 & -8 \\
1 & -4
\end{bmatrix}
\]
(b)
\[
A = \begin{bmatrix}
2 & 2 & 1 \\
0 & 1 & 2 \\
0 & 0 & -1
\end{bmatrix}
\]
(c)
\[
A = \begin{bmatrix}
1 & 0 & 0 \\
-2 & 1 & 3 \\
1 & 1 & -1
\end{bmatrix}
\]
(d)
\[
A = \begin{bmatrix}
1 & 2 & -1 \\
2 & 4 & -2 \\
3 & 6 & -3
\end{bmatrix}
\]
In these exercises, you will explore the process of diagonalizing matrices, which is useful for simplifying many matrix calculations and for understanding the underlying structure of linear transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc40ed8c4-e2aa-4488-bef1-34c2accebf46%2F217cfd09-b956-4f32-aa97-cf489443b622%2Fh01cik_processed.png&w=3840&q=75)
Transcribed Image Text:**Diagonalization of Matrices**
In each of the following problems, factor the matrix \( A \) into a product \( XDX^{-1} \), where \( D \) is a diagonal matrix.
(a)
\[
A = \begin{bmatrix}
2 & -8 \\
1 & -4
\end{bmatrix}
\]
(b)
\[
A = \begin{bmatrix}
2 & 2 & 1 \\
0 & 1 & 2 \\
0 & 0 & -1
\end{bmatrix}
\]
(c)
\[
A = \begin{bmatrix}
1 & 0 & 0 \\
-2 & 1 & 3 \\
1 & 1 & -1
\end{bmatrix}
\]
(d)
\[
A = \begin{bmatrix}
1 & 2 & -1 \\
2 & 4 & -2 \\
3 & 6 & -3
\end{bmatrix}
\]
In these exercises, you will explore the process of diagonalizing matrices, which is useful for simplifying many matrix calculations and for understanding the underlying structure of linear transformations.
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