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Problem 1. $A = egin{bmatrix} 1 & 2 \ -1 & 4 end{bmatrix}$ a) Diagonalize matrix A. For that find an invertible matrix P, it's inverse $P^{-1}$ and a diagonal matrix D such that $A = PDP^{-1}$. b) What are eigenvalues of the matrix $A^n$? c) Use diagonalization to calculate $A^n$. Hint:it is easy to calculate $egin{bmatrix} a & b \ c & d end{bmatrix} egin{bmatrix} lambda_1 & 0 \ 0 & lambda_2 end{bmatrix} = egin{bmatrix} lambda_1 a & lambda_2 b \ lambda_1 c & lambda_2 d end{bmatrix}$

          Problem 1.
$A = egin{bmatrix} 1 & 2 \ -1 & 4 end{bmatrix}$
a) Diagonalize matrix A. For that find an invertible matrix P, it's inverse $P^{-1}$ and a diagonal matrix D such that $A = PDP^{-1}$.
b) What are eigenvalues of the matrix $A^n$?
c) Use diagonalization to calculate $A^n$.
Hint:it is easy to calculate
$egin{bmatrix} a & b \ c & d end{bmatrix} egin{bmatrix} lambda_1 & 0 \ 0 & lambda_2 end{bmatrix} = egin{bmatrix} lambda_1 a & lambda_2 b \ lambda_1 c & lambda_2 d end{bmatrix}$

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Problem 1.
A = eginbmatrix 1 2 -1 4 endbmatrix
a) Diagonalize matrix A. For that find an invertible matrix P, it's inverse P^-1 and a diagonal matrix D such that A = PDP^-1.
b) What are eigenvalues of the matrix A^n?
c) Use diagonalization to calculate A^n.
Hint:it is easy to calculate
eginbmatrix a b c d endbmatrix eginbmatrix lambda1 0 0 lambda2 endbmatrix = eginbmatrix lambda1 a lambda2 b lambda1 c lambda2 d endbmatrix

Added by Reginald R.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Supreeta N

04/05/2023

Step 1

To do that, we need to solve the characteristic equation det(A - λI) = 0, where λ are the eigenvalues and I is the identity matrix. The matrix A is given by: A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} The characteristic equation is: det(A - λI) = Show more…

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Problem 1: A = [[1, 2], [-1, 4]] a) Diagonalize matrix A. For that find an invertible matrix P, it's inverse P-1 and a diagonal matrix D such that A = PDP-1. b) What are eigenvalues of the matrix An? c) Use diagonalization to calculate An. Hint:it is easy to calculate [[a, b], [c, d]] [[lambda1, 0], [0, lambda2]] = [[lambda1a, lambda2b], [lambda1c, lambda2d]]

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