APM3700 Assignment
3 (COMPLETE
ANSWERS) 2024 - DUE
28 August 2024 ; 100%
TRUSTED Complete,
trusted solutions and
explanations.
ADMIN
[COMPANY NAME]
, APM3700 Assignment 3 (COMPLETE ANSWERS) 2024 -
DUE 28 August 2024 ; 100% TRUSTED Complete, trusted
solutions and explanations.
QUESTION 1 If = 3 6 1 4 B , find the eigenvalues of B.
(5)
To find the eigenvalues of the matrix B=(3614)B =
\begin{pmatrix} 3 & 6 \\ 1 & 4 \end{pmatrix}B=(3164), we
follow these steps:
Step 1: Find the characteristic equation
The characteristic equation of a matrix BBB is given by:
det(B−λI)=0\text{det}(B - \lambda I) = 0det(B−λI)=0
where λ\lambdaλ is an eigenvalue and III is the identity matrix.
Given:
B=(3614),I=(1001)B = \begin{pmatrix} 3 & 6 \\ 1 & 4
\end{pmatrix}, \quad I = \begin{pmatrix} 1 & 0 \\ 0 & 1
\end{pmatrix}B=(3164),I=(1001)
So,
B−λI=(3614)−λ(1001)=(3−λ614−λ)B - \lambda I =
\begin{pmatrix} 3 & 6 \\ 1 & 4 \end{pmatrix} - \lambda
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} =
\begin{pmatrix} 3 - \lambda & 6 \\ 1 & 4 - \lambda
\end{pmatrix}B−λI=(3164)−λ(1001)=(3−λ164−λ)
Step 2: Compute the determinant