One of the eigenvalues of the matrix1(²2Cais 2(c+1), and a corresponding eigenvector isO-1-2c02 (2c+1)(c + 1)30-2 (2+ c)0(-²).What is the value of a?

Answered: One of the eigenvalues of the matrix 1…

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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Related questions

Question

One of the eigenvalues of the matrix
1
(²2
C
a
is 2(c+1), and a corresponding eigenvector is
O-1-2c
02 (2c+1)
(c + 1)
3
0-2 (2+ c)
0
(-²).
What is the value of a?

Expert Solution

Step 1

We have $A = (\begin{matrix} 1 & - a \\ c & a \end{matrix})$

Now, given that $λ = 2 (c + 1)$ and the corresponding eigenvector is $X = (\begin{matrix} - 2 \\ 1 \end{matrix})$

Consider:

$(A - λ I) X = 0 \Rightarrow (A - (2 (c + 1)) I) (\begin{matrix} - 2 \\ 1 \end{matrix}) = 0 \Rightarrow ((\begin{matrix} 1 & - a \\ c & a \end{matrix}) - (2 c + 2) (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})) (\begin{matrix} - 2 \\ 1 \end{matrix}) = 0 (A - (2 (c + 1)) I) (\begin{matrix} - 2 \\ 1 \end{matrix}) = 0 \Rightarrow ((\begin{matrix} 1 & - a \\ c & a \end{matrix}) - (\begin{matrix} 2 c + 2 & 0 \\ 0 & 2 c + 2 \end{matrix})) (\begin{matrix} - 2 \\ 1 \end{matrix}) = 0 \Rightarrow (\begin{matrix} 1 - 2 c - 2 & - a \\ c & a - 2 c - 2 \end{matrix}) (\begin{matrix} - 2 \\ 1 \end{matrix}) = 0 \Rightarrow (\begin{matrix} - 1 - 2 c & - a \\ c & a - 2 c - 2 \end{matrix}) (\begin{matrix} - 2 \\ 1 \end{matrix}) = 0 \Rightarrow (\begin{matrix} (- 1 - 2 c) (- 2) - a \\ - 2 c + a - 2 c - 2 \end{matrix}) = 0 \Rightarrow (\begin{matrix} 2 + 4 c - a \\ - 4 c + a - 2 \end{matrix}) = 0$