Find the Rank of a Matrix Using the Echelon Form of the above matrix.Give details step by step.
In: Advanced Math
Let \( B_1 = \left\{(2,1,1,1),(1,1,1,1),(1,1,2,1)\right\} \hspace{2mm} \)and \( \hspace{2mm}B_2 =\left\{(2,1,2,2)\right\}\hspace{2mm} \)be two subsets of\( \hspace{2mm} \mathbb{R}^4, E_1 \) be a subspace spanned by\( B_1, E_2 \)be a subspace spanned by \( B_2 \), and L be a linear operator on \( \mathbb{R}^4 \) defined by
\( L(v)=(-w +4x-y+z,-w+3x,-w+2x+y,-w+2x+z)\hspace{2mm},v=(w,x,y,z) \)
(a) Show that \( B_1 \) is a basis for \( E_1 \) and \( B_2 \) is a basis for \( E_2 \)
(b) Show that \( E_1 \) and \( E_2 \) are L-invariant. Find the matrices \( A_{1} =[L_{E_1}]_{B_1} \) and \( A_2=[L_{E_2}]_{B_2} \)
In: Advanced Math
Let A be a square matrix defined by \( A = \begin{pmatrix}6&2&3\\ -3&-1&-1\\ -5&-2&-2\end{pmatrix} \)L be a map from\( \hspace{2mm} \mathbb{R}^3\hspace{2mm} \)into\( \hspace{2mm}\mathbb{R}^3\hspace{2mm} \)by\( \hspace{2mm} L(v) = Av. \)
(a) Show that L is a linear operator on \( \hspace{2mm}\mathbb{R}^3. \)
(b) Find the characteristic polynomial of L with respect to standard basis for \( \mathbb{R}^3 \) Derive the determinant of L then deduce that L is invertible.
(c) Find the eigenvalues and eigenspaces of L.
(d) Show that L is not diagonalizable, but it is triangularizable, then triangularize L.
(e) Write \( L^n \) in term of n, where \( L^n = L(L(...(L)..)) \), the n compositions of L.
In: Advanced Math
Let A be a square matrix defined by \( A =\begin{pmatrix}-1&-2&-1&3\\ -6&-5&1&6\\ -6&-4&0&6\\ -6&-7&1&8\end{pmatrix} \) and its characteristics polynomial \( P(\lambda)=\bigg(\lambda+1\bigg)^2\bigg(\lambda-2\bigg)^2 \)
(a) Find the minimal polynomial of A.
(b) Deduce that A is not diagonalizable, but it is triangularizable, then triangularize A.
(c) Write \( A^n \) in terms of n.
In: Advanced Math
Let A be a square matrix defined by \( A = \begin{pmatrix}-2&-1&-5\\ 2&2&3\\ 4&2&2\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Find the eigenvalues and eigenspaces of A.
(c) Show that A is not diagonalizable, but it is triangularizable, then triangularize A.
(d) Find the three real sequences \( (a)_n, (b)_n ,(c)_n \) satisfying.
\( \begin{cases} a_{n+1}=-2a_n-b_n-5c_n \hspace{2mm},a_0=1 & \quad \\ b_{n+1}=2a_n+2b_n+3c_n \hspace{2mm}, b_0=0 & \quad \\ c_{n+1}=4a_n+2b_n+6c_n \hspace{2mm},c_0=1 & \quad \end{cases} \)
In: Advanced Math
Let A be a square matrix defined by \( A =\begin{pmatrix}-8&-3&-6\\ 4&0&4\\ 4&2&2\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Find the eigenvalues and eigenspaces of A.
(c) Show that A is not diagonalizable, but it is triangularizable, then triangularize A.
(d) Write \( A^n \) in terms of \( I, A,A^2 \) and n.
In: Advanced Math
Let A be a square matrix defined by \( A = \begin{pmatrix}-3&-1&-3\\ 5&2&5\\ -1&-1&-1\end{pmatrix} \)
(a) Find the characteristic polynomial of A.
(b) Find the eigenvalues of A. Show that A is not diagonalizable over \( \mathbb{R} \)
(c) Show that A is diagonalizable over\( \mathbb{C} \). Find the eigenspaces. Diagonalize A.
(d) Express \( A^n \) in the form of \( a_nA^2+b_ nA+c_nI_n \) where \( (a_n), (b_n) \) and \( (c_n) \) are real sequences to be specified.
\( A=PDP^{-1},D=\begin{pmatrix}0&0&0\\ 0&-1-i&0\\ 0&0&-1+i\end{pmatrix},P=\begin{pmatrix}-1&-1-2i&-1+2i\\ 0&1+3i&1-3i\\ 1&1&1\end{pmatrix} \)
In: Advanced Math
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