a. Let A be a square matrix with integer entries. Prove that if lambda is a...

a. Let A be a square matrix with integer entries. Prove that if lambda is a rational eigenvalue of A then in fact lambda is an integer.

b. Prove that the characteristic polynomial of the companion matrix of a monic polynomial f(t) equals f(t).

Expert Solution

Colby Messinger answered 1 year ago

(a) Let a > 1 be an integer. Prove that any composite divisor of a −...

(a) Let a > 1 be an integer. Prove that any composite divisor of a − 1 is a pseudoprime of base a. (b) Suppose, for some m, than n divides a^(m − 1) and n ≡ 1 (mod m). Prove that if n is composite, then n is a pseudoprime of base a. (c) Use (b) to give two examples pseudoprimes of base a with a = 2 and a = 3 (hint: take m = 2k to be...

Write a MIPS assembly language to transpose a square integer matrix in code

Give an example of a square matrix A such that all the diagonal entries are positive...

Give an example of a square matrix A such that all the diagonal entries are positive but A is not positive definite

Prove uniqueness of (a) LU-factorisation, (b) of LDU-factorisation of a square matrix.

Let E = Q(√a), where a is an integer that is not a perfect square. Show...

Let E = Q(√a), where a is an integer that is not a perfect square. Show that E/Q is normal

Given that the square matrix, A is nilpotent (Ak = 0 for some positive integer k)....

Given that the square matrix, A is nilpotent (Ak = 0 for some positive integer k). If A is n by n, show that An = 0.

Let A be a m × n matrix with entries in R. Recall that the row...

Let A be a m × n matrix with entries in R. Recall that the row rank of A means the dimension of the subspace in RN spanned by the rows of A (viewed as vectors in Rn), and the column rank means that of the subspace in Rm spanned by the columns of A (viewed as vectors in Rm). (a) Prove that n = (column rank of A) + dim S, where the set S is the solution space...

8.Let a and b be integers and d a positive integer. (a) Prove that if d...

8.Let a and b be integers and d a positive integer. (a) Prove that if d divides a and d divides b, then d divides both a + b and a − b. (b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing that the converse is false. 9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c,...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties

Let n be a positive integer. Prove that if n is composite, then n has a...

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

Question