b) Consider the matrix A = −2 10 1 0 1 0 4 20 1 . (i) Compute the characteristic polynomial det(xI −A) of A. Hence or otherwise show that the eigenvalues of A are \lambda = 1, 2, −3.
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Given matrix \( A \): \[ A = \begin{pmatrix} -2 & 10 & 1 \\ 0 & 1 & 0 \\ 4 & 20 & 1 \end{pmatrix} \] Show more…
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Let $A=\left[a_{i j}\right.$ be an $n \times n$ matrix. The matrix $A-\lambda I$ is called the characteristics matrix of $A$, where $\lambda$ is a scalar and $I$ is the identity matrix. The determinant $|A-\lambda I|$ is a non-null polynomial of degree $n$ in $\lambda$ and is called the characteristic polynomial of $A$. The equation $|A-\lambda I|=0$ is called the characteristic equation of $A$ and its roots are called the characteristic roots or latent roots or eigen values of $A$. The set of all eigenvalues of the matrix $A$ is called the spectrum of A. The product of the eigenvalues of a matrix $A$ is equal to the determinant $A$. The given values of the matrix $A=\left[\begin{array}{lll}1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4\end{array}\right]$ are (A) $4,-2,-2$, (B) $-4,2,-2$ (C) $-4,2,2$ (D) $4,-4,2$
Let $A=\left[a_{i j}\right.$ be an $n \times n$ matrix. The matrix $A-\lambda I$ is called the characteristics matrix of $A$, where $\lambda$ is a scalar and $I$ is the identity matrix. The determinant $|A-\lambda I|$ is a non-null polynomial of degree $n$ in $\lambda$ and is called the characteristic polynomial of $A$. The equation $|A-\lambda I|=0$ is called the characteristic equation of $A$ and its roots are called the characteristic roots or latent roots or eigen values of $A$. The set of all eigenvalues of the matrix $A$ is called the spectrum of A. The product of the eigenvalues of a matrix $A$ is equal to the determinant $A$. $\left.\begin{array}{l}\text { The characteristic roots of the matrix } A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 2 \\ 1 & 2 & 0\end{array}\right]\end{array}\right]$ (A) 1 (B) 2 (C) $-2$ (D) 3
7. It is known that matrix A has the following eigenvalues -2+5i, -2-5i. a) Find the characteristic equation that has the roots above. b) Create a matrix A satisfying the condition above, with all nonzero, real, and distinct entries a, b, c, and d. c) Find the inverse of your own matrix A. Confirm that matrix A satisfies its own characteristic equation (Cayley-Hamilton Theorem).
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