, 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: Set up the characteristic equation
The characteristic equation is given by:
det(B−λI)=0\det(B - \lambda I) = 0det(B−λI)=0
where III is the identity matrix and λ\lambdaλ represents the
eigenvalues.
Step 2: Compute B−λIB - \lambda IB−λI
Subtract λI\lambda IλI from the matrix 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 3: Calculate the determinant
Compute the determinant of the matrix B−λIB - \lambda IB−λI:
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: Set up the characteristic equation
The characteristic equation is given by:
det(B−λI)=0\det(B - \lambda I) = 0det(B−λI)=0
where III is the identity matrix and λ\lambdaλ represents the
eigenvalues.
Step 2: Compute B−λIB - \lambda IB−λI
Subtract λI\lambda IλI from the matrix 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 3: Calculate the determinant
Compute the determinant of the matrix B−λIB - \lambda IB−λI:
