Question

Let A be an n×n matrix. If there exists k > n such that A^k =0,then...

Let A be an n×n matrix. If there exists k > n such that A^k =0,then
(a) prove that In − A is nonsingular, where In is the n × n identity matrix;
(b) show that there exists r ≤ n such that A^r= 0.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Show that for each nonsingular n x n matrix A there exists a permutation matrix P...
Show that for each nonsingular n x n matrix A there exists a permutation matrix P such that P A has an LR decomposition. [note: A factorization of a matrix A into a product A=LR of a lower (left) triangular matrix L and an upper (right) triangular matrix R is called an LR decomposition of A.]
Let A be an n×n nonsingular matrix. Denote by adj(A) the adjugate matrix of A. Prove:...
Let A be an n×n nonsingular matrix. Denote by adj(A) the adjugate matrix of A. Prove: 1)   det(adj(A)) = (det(A)) 2)    adj(adj(A)) = (det(A))n−2A
Let x1, x2, ..., xk be linearly independent vectors in R n and let A be...
Let x1, x2, ..., xk be linearly independent vectors in R n and let A be a nonsingular n × n matrix. Define yi = Axi for i = 1, 2, ..., k. Show that y1, y2, ..., yk are linearly independent.
Exercise1.2.1: Prove that if t > 0 (t∈R), then there exists an n∈N such that 1/n^2...
Exercise1.2.1: Prove that if t > 0 (t∈R), then there exists an n∈N such that 1/n^2 < t. Exercise1.2.2: Prove that if t ≥ 0(t∈R), then there exists an n∈N such that n−1≤ t < n. Exercise1.2.8: Show that for any two real numbers x and y such that x < y, there exists an irrational number s such that x < s < y. Hint: Apply the density of Q to x/(√2) and y/(√2).
Let A be an m × n matrix, and Q be an n × n invertible...
Let A be an m × n matrix, and Q be an n × n invertible matrix. (1) Show that R(A) = R(AQ), and use this result to show that rank(AQ) = rank(A); (2) Show that rank(AQ) = rank(A).
Let A be a 2 × 2 matrix satisfying A^k = 0 for some positive integer...
Let A be a 2 × 2 matrix satisfying A^k = 0 for some positive integer k. Show that A^2 = 0.
Let R*= R\ {0} be the set of nonzero real numbers. Let G= {2x2 matrix: row...
Let R*= R\ {0} be the set of nonzero real numbers. Let G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in R*, b in R} (a) Prove that G is a subgroup of GL(2,R) (b) Prove that G is Abelian
Let A be an n x M matrix and let T(x) =A(x). Prove that T: R^m...
Let A be an n x M matrix and let T(x) =A(x). Prove that T: R^m R^n is a linear transformation
(a) Consider binary codes determined by a parity-check matrix H. Let r be a vector of...
(a) Consider binary codes determined by a parity-check matrix H. Let r be a vector of received symbols. The syndrome of r happens to be a sum of some columns of H. What columns are these? Hint: They are determined by the errors occurring in transmissions. (b) Let G be a k × n generating matrix of a code C the form G = [I_k | B] , where Ik is the k × k identity matrix, and B is...
Let Zn = {0, 1, 2, . . . , n − 1}, let · represent...
Let Zn = {0, 1, 2, . . . , n − 1}, let · represent multiplication (mod n), and let a ∈ Zn. Prove that there exists b ∈ Zn such that a · b = 1 if and only if gcd(a, n) = 1.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT