Let M in R^{10×10}, s.t. M^{2020} =0. Prove M^{10} = 0

asked 2020-10-19
Let M \in R^{10×10}, s.t. M^{2020} =0. Prove M^{10} = 0

Answers (1)

Let A10×10 matrix. Given that A^{2020}=0. Let m(x) be the minimal polynomial of A. By Cayley-Hamilton, the degree of mm is atmost 10. By the propert of minimal polynomial m(x) devides x2020. Then the possibilty of are m(x)=x^{k} for some k=1,2,⋯ ,10.
Case 1: If $k=10, $ and since A satisfies its minial polynomial we have A10=0.
Case 2: If k\(A^{10}=A^{10}-kA^{k}=A10-k\times0=0.\)
In Any case we have A^{10}=0.

Relevant Questions

asked 2020-11-12
Prove that the following determinant is equal to 0 :
asked 2021-03-12
Let a linear sytem of equations Ax=b where
\(A=\begin{pmatrix}4 & 2&-2 \\2 & 2&-3\\-2&-3&14 \end{pmatrix} , b=\begin{pmatrix}10 , 5 , 4 \end{pmatrix}^T\)
in case we solve this equation system by using Dolittle LU factorization method , find Z and X matrices
asked 2021-02-14
Prove the following
A(rB)= r(AB)= (rA)B
where r and s are real numbers and A and B are matrices
asked 2020-12-02
Let A and B be \(n \times n\) matrices. Recall that the trace of A , written tr(A),equal
Prove that tr(AB)=tr(BA) and \(tr(A)=tr(A^t)\)
asked 2021-02-14
Prove that if A has rank r with r > 0, then A has an r×r invertible submatrix.
asked 2021-03-06
Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer. The eigenvalues of AB are all real.
asked 2020-12-05
Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1),(1, 1),(1, 2),(2, 0),(2, 2),(3, 0). Find reflexive, symmetric and transitive closure of R.
asked 2020-12-16
Compute the following
a) \begin{bmatrix}-5 & -4&3&-10&-3&6 \\6&-10&5&9&4&-1 \end{bmatrix}+\begin{bmatrix}-7 & 3&10&0&8&8 \\8&0&4&-3&-8&0 \end{bmatrix}
b) -5\begin{bmatrix}8 & -10&7 \\0 & -9&7\\10&-5&-10\\1&5&-10 \end{bmatrix}
c)\begin{bmatrix}3 & 0&-8 \\6 & -4&-2\\6&0&-8\\-9&-7&-7 \end{bmatrix}^T
asked 2021-01-16
Let W be the vector space of \(3 \times 3\) symmetric matrices , \(A \in W\) Then , which of the following is true ?
a) \(A^T=1\)
b) \(dimW=6\)
c) \(A^{-1}=A\)
d) \(A^{-1}=A^T\)
asked 2021-03-18
Let A and B be similar nxn matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if \(X^2=X\) )