Determining Value of a in Matrix A with $\lambda$ = 0

In summary, the conversation discusses finding the value of a given that $\lambda=0$ is an eigenvalue of matrix A. The participant suggests using the characteristic polynomial and the determinant to solve for a. Another participant mentions that the trace can also be used to find the value of a. The final conclusion is that a=3 is the correct solution.
  • #1
Yankel
395
0
Hello all,

Given the following matrix,

\[A=\begin{pmatrix} 2 & 6\\ 1 & a \end{pmatrix}\]

and given that

\[\lambda =0\]

is an eigenvalue of A, I am trying to determine that value of a.

What I did, is to create the characteristic polynomial

\[(\lambda -2)*(\lambda -a)+6=0\]

and given

\[\lambda =0\]

I got that a is -3.

Somehow I am not sure. Is there a way of finding the second eigenvalue before calculating a ?

Thank you !
 
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  • #2
You could use that the determinant is the product of the eigenvalues.
(Also, the trace is the sum of the eigenvalues, but in this case you do not need that to determine $a$. You could use it to determine the second eigenvalue, though.)
 
Last edited:
  • #3
Yankel said:
Hello all,

Given the following matrix,

\[A=\begin{pmatrix} 2 & 6\\ 1 & a \end{pmatrix}\]

and given that

\[\lambda =0\]

is an eigenvalue of A, I am trying to determine that value of a.

What I did, is to create the characteristic polynomial

\[(\lambda -2)*(\lambda -a)+6=0\]

and given

\[\lambda =0\]

I got that a is -3.

Somehow I am not sure. Is there a way of finding the second eigenvalue before calculating a ?

Thank you !
Check the signs in that calculation! I think that it should be $-6$ in the characteristic polynomial.
 
  • #4
Opalg, you are correct, it is -6 and therefore a=3. Am I correct ?
 
  • #5
Yankel said:
Somehow I am not sure. Is there a way of finding the second eigenvalue before calculating a ?

As Krylov pointed out, the trace is the sum of the eigenvalues.
So if one eigenvalue is 0, then the other is $\operatorname{Tr}A=2+a$.

And yes, a=3 is the correct solution.
 

Related to Determining Value of a in Matrix A with $\lambda$ = 0

1. How do you determine the value of a in a matrix A with lambda = 0?

To determine the value of a in a matrix A with lambda = 0, you would need to use the characteristic equation, which is det(A - λI) = 0. By substituting 0 for λ, you can solve for the value of a.

2. What is the purpose of determining the value of a in a matrix A with lambda = 0?

The value of a in a matrix A with lambda = 0 helps to determine the eigenvectors of the matrix. This is important in many applications, such as finding the steady state of a system or analyzing the behavior of linear transformations.

3. Can the value of a in a matrix A with lambda = 0 be negative?

Yes, the value of a in a matrix A with lambda = 0 can be negative. This depends on the specific matrix and its coefficients. It is important to remember that lambda is just a variable and does not have a specific value until it is substituted into the characteristic equation.

4. Is there a specific method for determining the value of a in a matrix A with lambda = 0?

Yes, there are multiple methods for determining the value of a in a matrix A with lambda = 0. Some common methods include using the characteristic equation and solving for a, using the trace and determinant of the matrix, or using the Cayley-Hamilton theorem.

5. How does the value of a in a matrix A with lambda = 0 affect the eigenvalues of the matrix?

The value of a in a matrix A with lambda = 0 does not directly affect the eigenvalues of the matrix. However, it does play a role in determining the eigenvectors and the corresponding eigenvalues. The value of a helps to determine the characteristic equation, which is used to solve for the eigenvalues.

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