Let A be a square matrix defined by \( A =\begin{pmatrix}-1&3&-1\\ -3&5&-1\\ -3&3&1\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Show that A is diagonalizable then diagonalize it.
(c) Write \( A^n \) in term of n.
In: Advanced Math
Let A be a square matrix defined by\( A =\begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Show that A is diagonalizable then diagonalize it.
(c) Write $A^n$ \hspace{2mm} in term of n.
In: Advanced Math
Let A be a square matrix defined by \( A = \begin{pmatrix}3&2\\ 3&-2\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Show that A is diagonalizable then diagonalize it.
(c) Write \( A^n \) in term of n.
In: Advanced Math
Determine the value of a so that \( \lambda = 2 \) is an eigenvalue of
\( A=\begin{pmatrix}1&-1&0\\ a&1&1\\ 0&1+a&3\end{pmatrix} \)
Then show that A is diagonallizable and diagonalize it.
In: Advanced Math
Let A be a square matrix defined by \( A=\begin{pmatrix}4&-2&1\\ 2&0&1\\ 2&-2&3\end{pmatrix}\hspace{2mm} \)Find the minimal polynomial of A. Then express \( A^4 \) and \( A^{-1} \) in terms of A and I.
In: Advanced Math
Let \( A\in M_n(\mathbb{R})\hspace{2mm} \) and \( \hspace{2mm}\lambda_1, \lambda_2,...,\lambda_n \hspace{2mm} \)(no need distinct) be eigenvalues of A. Show that
a). \( \sum _{i=1}^n\lambda _i=tr\left(A\right) \) b). \( \:\prod _{i=1}^n\lambda _i=\left|A\right|\: \)
In: Advanced Math
Let \( A\in M_6(\mathbb{R}) \) be an invertible matrix satisfies \( A^3-4A^2 + 3A = 0 \) and \( tr(A) = 8. \) Find the characteristics polynomial of A.
In: Advanced Math
Let \( A\in M_n(\mathbb{R})\hspace{2mm} \) and\( \hspace{2mm} m_A(\lambda)\hspace{2mm} \) be its minimal polynomial. Let f be a polynomial satisfies\( \hspace{2mm}f(A) = 0. \hspace{2mm} \)Show that\( \hspace{2mm} f(\lambda) \hspace{2mm} \)is divisible by\( \hspace{2mm} m_A(\lambda). \)
In: Advanced Math
Determine all possible Jordan canonical forms J for a matrix of order 6 whose minimal polynomial is
\( m(\lambda)=\bigg(\lambda-1\bigg)^3\bigg(\lambda-3\bigg)^2 \)
In: Advanced Math
Determine all possible Jordan canonical forms for a linear operator L whose characteristic polynomial is
\( P(\lambda)=\bigg(\lambda-3\bigg)^3\bigg(\lambda-4\bigg)^2 \)
In: Advanced Math
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L.
B=\( \begin{pmatrix}2&1&0&0\\ 0&2&0&0\\ 0&0&3&0\\ 0&0&0&3\end{pmatrix} \)
In: Advanced Math
The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L.
A=\( \begin{pmatrix}2&0&0&0\\ 0&2&0&0\\ 0&0&1&0\\ 0&0&0&3\end{pmatrix} \)
In: Advanced Math
Find the characteristics and the minimal polynomial of the following matrices over R, then deduce their corresponding Jordan Canonical Form J.
C=\( \begin{pmatrix}1&1&0&0\\ -1&-1&1&0\\ 0&1&1&0\\ -1&-1&1&1\end{pmatrix} \)
In: Advanced Math
Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J.
B=\( \begin{pmatrix}2&-1&-1&2\\ 0&1&-1&2\\ 2&-5&-1&6\\ 1&-3&-2&6\end{pmatrix} \)
In: Advanced Math
Find the characteristics and the minimal polynomial of the following matrices over \( \mathbb{R} \), then deduce the their corresponding Jordan Canonical Form J.
A=\( \begin{pmatrix}1&-1&-1\\ 0&0&-1\\ 0&1&2\end{pmatrix}\: \)
In: Advanced Math