You are given the 2x2 matrix M= (k 3) , where k is not 2.
(0 2)
i)Find the eigenvalues of M, and the corresponding eigenvectos.
ii)Express M in the form UDU^(-1), where D is a diagonal matrix.
iii)Hence find the matrix M^n.
k 3
0 2
thats the 2x2 matrix, i m really stuck
det [ A - L I] = 0
det [ k-L , 3 / 0 , 2-L ] = 0
(k-L)(2-L) - 3*0 = 0
2k - (2+k) L + L^2= 0
L = {(2+k) +/- sqrt (k^2-4k+4) } / 2 but sqrt(k^2-4k+4) = (k-2)
so
L = {k+2 +k-2}/2 = k
or
L = {k+2 -k+2}/2 = 2
eigenvalues are k and 2
suggest this:
https://www.youtube.com/watch?v=IdsV0RaC9jM
To find the eigenvalues and eigenvectors of a 2x2 matrix M, we need to follow these steps:
i) Find the eigenvalues:
The eigenvalues of a matrix can be found by solving the characteristic equation. The characteristic equation is obtained by subtracting λ (the eigenvalue) from the diagonal elements of the matrix and then calculating the determinant, setting it equal to zero.
For the given matrix M:
| k-λ 3 |
| 0 2-λ |
The determinant of M can be calculated as (k-λ)(2-λ) - (3*0) = (k-λ)(2-λ) = 0.
Setting it equal to zero, we can solve for the eigenvalues λ:
(k-λ)(2-λ) = 0
Expanding the above equation, we get:
2k - kλ - 2λ + λ^2 = 0
Rearranging the terms, we have:
λ^2 - (k+2)λ + 2k = 0
We can solve this quadratic equation to find the eigenvalues.
ii) Find the eigenvectors:
Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the system of equations (M - λI)x = 0, where λ is the eigenvalue, M is the given matrix, and I is the identity matrix.
To find the matrix M in the form UDU^(-1), where D is a diagonal matrix, we need to find the matrix U consisting of eigenvectors of M. U is a matrix whose columns are the eigenvectors.
The matrix D is a diagonal matrix with the eigenvalues as its diagonal entries.
iii) To find the matrix M^n, we can use the formula M^n = U * D^n * U^(-1), where U and D are as defined above.
By substituting the values of U, D, and n into the formula, we can calculate M^n.