× Close
Join the Ques10 Community
Ques10 is a community of thousands of students, teachers, and academic experts, just like you.
Join them; it only takes a minute
Sign up
Question: If A = $\begin{bmatrix} 1&4 \\ 2&3 \end{bmatrix}$. Find A$^{50}$
0

Subject: Applied Mathematics 4

Topic: Matrix Theory

Difficulty: Medium

m4e(34) • 95 views
ADD COMMENTlink
modified 19 days ago  • written 3 months ago by gravatar for Manan Bothra Manan Bothra0
0

For characteristic equation |A - $\lambda$ I| = 0

$$ \begin{vmatrix} 1 - \lambda & 4 \\ 2 & 3 - \lambda \end{vmatrix} = 0 $$

$ (1 - \lambda)(3 - \lambda) - 8 = 0 \\ 3 - \lambda - 3\lambda + \lambda^2 - 8 = 0 \\ \lambda^2 - 5\lambda + \lambda - 5 = 0 \\ (\lambda + 1)(\lambda - 5) = 0 \lambda = -1,5 $

Let,

$ \phi(A) = A^{50} \\ \phi(A) = aA + bI \\ \phi(\lambda) = \lambda^{50} \\ \phi(\lambda) = a\lambda + b \\ \phi(-1) = 1 \\ \phi(5) = 5^{50} $

case(i) $\lambda$ = -1

$ \phi(-1) = a(-1) + b \\ a - b = 1 \hspace{0.25cm} ...(1) $

case(ii) $\lambda$ = 5

$ \phi(5) = a(5) + b \\ 5^{50} = 5a + b \\ 5a + b = 5^{50} \hspace{0.25cm} ...(2) \\ 6a = 5^{50} - 1 \hspace{0.5cm} [From \,\, (1)] \\ a = \frac{5^{50} - 1}{6} \therefore b = \frac{5^{50} + 5}{6} $

$ \phi(A) = aA + bI \\ = \frac{5^{50} - 1}{6} \begin{bmatrix} 1&4 \\ 2&3 \end{bmatrix} + \frac{5^{50} + 5}{6} \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} \\ = \begin{bmatrix} \frac{5^{50} - 1}{6} & 4(\frac{5^{50} - 1}{6}) \\ 2(\frac{5^{50} - 1}{6}) & 3(\frac{5^{50} - 1}{6}) \end{bmatrix} + \begin{bmatrix} \frac{5^{50} + 5}{6} & 0 \\ 0 & \frac{5^{50} + 5}{6} \end{bmatrix} \\ = \begin{bmatrix} 2(\frac{5^{50} +2}{6}) & 4(\frac{5^{50} - 1}{6}) \\ 2(\frac{5^{50} - 1}{6}) & \frac{4 \times 5^{50} + 2}{6} \end{bmatrix} \\ = \begin{bmatrix} \frac{5^{50} + 2}{3} & 2(\frac{5^{50} - 1}{3}) \\ \frac{5^{50} - 1}{3} & \frac{2 \times 5^{50} + 1}{3} \end{bmatrix} $

ADD COMMENTlink
written 19 days ago by gravatar for Manan Bothra Manan Bothra0
Please log in to add an answer.


Use of this site constitutes acceptance of our User Agreement and Privacy Policy.