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MAT3700 Assignment 3 (COMPLETE ANSWERS)2024 - DUE 28 August 2024 ; 100% TRUSTED Complete, trusted solutions and explanations. $2.50   Add to cart
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MAT3700 Assignment 3 (COMPLETE ANSWERS)2024 - DUE 28 August 2024 ; 100% TRUSTED Complete, trusted solutions and explanations.

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  • Mathematics III
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  • University Of South Africa
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  • Engineering Mathematics-II

MAT3700 Assignment 3 (COMPLETE ANSWERS)2024 - DUE 28 August 2024 ; 100% TRUSTED Complete, trusted solutions and explanations.

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  • August 22, 2024
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, 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, where:
3 & 6 \\ 1 & 4 \end{pmatrix} \] we need to solve the
characteristic equation, which is given by: \[ \text{det}(B -
\lambda I) = 0 \] where \( \lambda \) is the eigenvalue and \( I \)
is the identity matrix of the same size as \( B \). 1. **Subtract
\(\lambda\) times the identity matrix \( I \) from \( B \):** \[ B -
\lambda I = \begin{pmatrix} 3 & 6 \\ 1 & 4 \end{pmatrix} -
\lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} =
\begin{pmatrix} 3 - \lambda & 6 \\ 1 & 4 - \lambda
\end{pmatrix} \] 2. **Calculate the determinant of \( B -
\lambda I \):** \[ \text{det}(B - \lambda I) = \text{det}
\begin{pmatrix} 3 - \lambda & 6 \\ 1 & 4 - \lambda
\end{pmatrix}
The determinant of a 2x2 matrix (abcd)\begin{pmatrix} a & b \\
c & d \end{pmatrix}(acbd) is calculated as:
det=ad−bc\text{det} = ad - bcdet=ad−bc
Applying this to our matrix:
det(B−λI)=(3−λ)(4−λ)−(6⋅1)\text{det}(B - \lambda I) = (3 -
\lambda)(4 - \lambda) - (6 \cdot 1)det(B−λI)=(3−λ)(4−λ)−(6⋅1)

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