APM3700 Assignment 3 (COMPLETE ANSWERS) 2024 - DUE 28 August 2024
APM3700 Assignment 3 (COMPLETE ANSWERS) 2024 - DUE 28 August 2024
APM3700 Assignment 3 (COMPLETE ANSWERS) 2024 - DUE 28 August 2024 ; 100% TRUSTED Complete, trusted solutions and explanations.
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University of South Africa
Differential Equations
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, 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 eigenvalues λ\lambdaλ of a matrix BBB are found by
solving the characteristic equation:
det(B−λI)=0\text{det}(B - \lambda I) = 0det(B−λI)=0
where III is the identity matrix of the same size as BBB.
For matrix B=(3614)B = \begin{pmatrix} 3 & 6 \\ 1 & 4
\end{pmatrix}B=(3164), the matrix B−λIB - \lambda IB−λI is:
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: Calculate the determinant of B−λIB - \lambda IB−λI
The determinant of B−λIB - \lambda IB−λI is:
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