Math Help - Homogenous system of linear equations.

1. Homogenous system of linear equations.

So I've been doing this for a VERY long time now I don't seem to be getting the right answer (according to online solvers) I'm wondering if you guys can spot my error because this is REALLY frustrating.
$A=\left(\begin{matrix} 2/5&1/5\\3 &0\end{matrix}\right)$
And I found one of the Eigenvalues to be 1. so now subtracting the Diagonal from 1 I get this and I start to solve the homogeneous system of equation.
$(L=1)=\left(\begin{matrix} 3/5&1/5\\3 &1\end{matrix}\right)$ = $\left(\begin{matrix} 3&1\\3 &1\end{matrix}\right)$ = $\left(\begin{matrix} 1&1/3\\0 &0\end{matrix}\right)$
Now from here x(2)=T and x(1)+(1/3)T=0 --->X(1)=(-1/3)T
SO $X=\left(\begin{matrix} -1\\3\end{matrix}\right)$ BUT according toe the online solver it should be $X=\left(\begin{matrix} 1\\3\end{matrix}\right)$
I have tried everything so I need someone's help, WHAT AM I DOING WRONG?
Thank you!

2. Re: Homogenous system of linear equations.

Originally Posted by calculuskid1
So I've been doing this for a VERY long time now I don't seem to be getting the right answer (according to online solvers) I'm wondering if you guys can spot my error because this is REALLY frustrating.
$A=\left(\begin{matrix} 2/5&1/5\\3 &0\end{matrix}\right)$
And I found one of the Eigenvalues to be 1. so now subtracting the Diagonal from 1 I get this and I start to solve the homogeneous system of equation.
$(L=1)=\left(\begin{matrix} 3/5&1/5\\3 &1\end{matrix}\right)$ = $\left(\begin{matrix} 3&1\\3 &1\end{matrix}\right)$ = $\left(\begin{matrix} 1&1/3\\0 &0\end{matrix}\right)$
Now from here x(2)=T and x(1)+(1/3)T=0 --->X(1)=(-1/3)T
SO $X=\left(\begin{matrix} -1\\3\end{matrix}\right)$ BUT according toe the online solver it should be $X=\left(\begin{matrix} 1\\3\end{matrix}\right)$
I have tried everything so I need someone's help, WHAT AM I DOING WRONG?
Thank you!
Please post the original question exactly as it's worded.

3. Re: Homogenous system of linear equations.

Consider a female bird population with two age groups: adults and juveniles. The birds remain juvenile for one year and then become adults. The adult survival rate is 2/5 (that is, 2/5 of the adult birds survive from one year to the next), the juvenile survival rate is 1/5 (that is, the proportion of juveniles still alive — and grown up — the next year), and the reproduction rate is 3 (that is, each adult bird will have 3 oﬀspring the following year).
a0 = 56, and j0 = 40. Find an exact expression for a{k} and {jk} (as functions
of k).
So I have my A as I stated before, and I know how to do it, I am just having huge problems trying to find the P matrix to complete the problem.

4. Re: Homogenous system of linear equations.

are you trying to find a P that diagonalizes A?

in any case, your original matrix is in error. the equation I-A = 0 is:

$\begin{bmatrix}\frac{3}{5}&\frac{-1}{5}\\-3&1 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}$

which leads to $x - \frac{1}{3}y = 0$ , so an eigenvector is (1,3).