


9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent ...
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
(c) If A is a square matrix and A2 = 0,then A = 0. (d) Let A, B be two square matrices. If (A + B) 2 = A2 + 2AB + B2 , then AB = BA.
4. Let A be a square matrix. Assume that Ak = 0 for some positive integer k. Then prove that a) 1-A is is invertible b) (1 - A)-1 = 1 + A + A + A + ................ + Ak-1 (This question is printed wrong in the text book, 10th edition. If you have this book, correct it)
5. Let AE Maxn(C). Recall that A is said to be nilpo tent if there exists a positive integer k such that A 0. Prove the following statements (a) If A is nilpotent, then A 0. (Hint: First show that if A is nilpotent, then the Jordan form of A is also nilpotent.) (b) If A is nilpotent, then tr(A) 0 (e) A is nilpotent if and only if the characteristic polynomial of A is (-1)"" (d) If A is...
Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant.
Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant.
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
Algebra We know that matrix multiplication is not commutative: if A and B are square matrices of the same size, AB and BA are usually different We say that A and B commute if it so happens that AB BA. Determine all numbers a, b, e, d, such that the matrix com- mutes with both Calculus An object with mass m is dragged along a horizontal plane by a force acting along a rope attached to the object as shown...
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and let A be an n x n skew-symmetric matrix. Simplify A(A^2(BA)^-1)^T b. Let A be a square matrix such that A^3 = 0. A is then called a nilpotent matrix. Define another matrix B by the expression B = I - A; Show that B is invertible and that its inverse is I + A + A^2 c. Let B = (-2,0,0 ; 0,0,0...
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3. An n x n matrix N is called nilpotent if Nd = 0 for some d. (a) Show that if N is nilpotent, then N" = 0. (b) Show that the dim ker N is the number of Jordan blocks in its Jordan canonical form (c) How many similarity classes of 5 x 5 nilpotent matrices are there?