Question

# Solve the given initial-value problem.$\mathbf{X}^{\prime}=\left(\begin{array}{lll}{0} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {0}\end{array}\right) \mathbf{X}, \quad \mathbf{X}(0)=\left(\begin{array}{l}{1} \\ {2} \\ {5}\end{array}\right)$

Solution

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Determine the eigenvalues of the matrix $\mathbf{A}$.

\begin{align*} \det\left(\mathbf{A}-\lambda\mathbf{I}\right) &= 0\\ \begin{vmatrix} -\lambda & 0 & 1\\ 0 & 1-\lambda& 0\\ 1 & 0 & -\lambda\\ \end{vmatrix} &= 0\\ -\lambda\left[\left(1-\lambda\right)\left(-\lambda\right)-0\right]-0\left(0-0\right)+1\left[0-\left(1-\lambda\right)\right]&=0\\ -\lambda\left(-\lambda+\lambda^2\right)-0+\left(-1+\lambda\right)&=0\\ \lambda^2-\lambda^3-1+\lambda&=0\\ -\lambda^3+\lambda^2+\lambda-1&=0\\ \lambda^3-\lambda^2-\lambda+1&=0\\ \left(\lambda-1\right)\left(\lambda^2-1\right)&=0\\ \left(\lambda-1\right)\left(\lambda-1\right)\left(\lambda+1\right)&=0\\ \left(\lambda+1\right)\left(\lambda-1\right)^2&=0\\ \lambda_1=-1\;\;\text{or}\;\;\lambda_2=1\;\;\text{or}\;\;\lambda_3&=1 \end{align*}

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