, 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 a matrix BBB, you need to solve the
characteristic equation:
det(B−λI)=0\text{det}(B - \lambda I) = 0det(B−λI)=0
where λ\lambdaλ represents the eigenvalues, III is the identity
matrix of the same size as BBB, and det\text{det}det stands for
the determinant.
Given the matrix BBB:
B=(3614)B = \begin{pmatrix} 3 & 6 \\ 1 & 4
\end{pmatrix}B=(3164)
Step 1: Subtract λ\lambdaλ times the identity matrix III
from BBB:
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: Find the determinant of B−λIB - \lambda IB−λI: