, MAT3700 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 BBB, follow these steps:
1. Define the matrix BBB:
B=(3614)B = \begin{pmatrix} 3 & 6 \\ 1 & 4
\end{pmatrix}B=(3164)
2. Find the characteristic polynomial:
The eigenvalues λ\lambdaλ are found by solving the
characteristic equation, which is obtained by finding the
determinant of B−λIB - \lambda IB−λI, where III is the
identity matrix.
The matrix B−λIB - \lambda IB−λI is:
B−λI=(3−λ614−λ)B - \lambda I = \begin{pmatrix} 3 -
\lambda & 6 \\ 1 & 4 - \lambda \end{pmatrix}B−λI=(3−λ1
64−λ)
To find the determinant:
det(B−λI)=∣3−λ614−λ∣\text{det}(B - \lambda I) =
\begin{vmatrix} 3 - \lambda & 6 \\ 1 & 4 - \lambda
\end{vmatrix}det(B−λI)=3−λ164−λ