Question

In: Advanced Math

Let A be an n × n matrix which is not 0 but A2 = 0....

Let A be an n × n matrix which is not 0 but A2 = 0. Let I be the identity matrix.

a)Show that A is not diagonalizable.

b)Show that A is not invertible.

c)Show that I-A is invertible and find its inverse.

Solutions

Expert Solution


Related Solutions

Let A be an n x n matrix satisfying A2=A (idempotent). Find all eigenvalues and eigenvectors...
Let A be an n x n matrix satisfying A2=A (idempotent). Find all eigenvalues and eigenvectors of A. I know that the eigenvalues are 0 and 1 -- I do not know how to find the eigenvectors.
Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0....
Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0. (b) Show that A is symmetric if and only if A^2= AA^T
Let A be a diagonalizable n × n matrix and let P be an invertible n...
Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6 0 −4 , A5 A5 =
Let A be a diagonalizable n × n matrix and let P be an invertible n...
Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find A5 A = 4 0 −4 5 −1 −4 6 0 −6
Let A be some m*n matrix. Consider the set S = {z : Az = 0}....
Let A be some m*n matrix. Consider the set S = {z : Az = 0}. First show that this is a vector space. Now show that n = p+q where p = rank(A) and q = dim(S). Here is how to do it. Let the vectors x1, . . . , xp be such that Ax1, . . . ,Axp form a basis of the column space of A (thus each x can be chosen to be some unit...
Let A be an m × n matrix and B be an m × p matrix....
Let A be an m × n matrix and B be an m × p matrix. Let C =[A | B] be an m×(n + p) matrix. (a) Show that R(C) = R(A) + R(B), where R(·) denotes the range of a matrix. (b) Show that rank(C) = rank(A) + rank(B)−dim(R(A)∩R(B)).
Let v1 be an eigenvector of an n×n matrix A corresponding to λ1, and let v2,...
Let v1 be an eigenvector of an n×n matrix A corresponding to λ1, and let v2, v3 be two linearly independent eigenvectors of A corresponding to λ2, where λ1 is not equal to λ2. Show that v1, v2, v3 are linearly independent.
Let 3x3 matrix A = -3 0 -4                               0 5 0        &nb
Let 3x3 matrix A = -3 0 -4                               0 5 0                              -4 0 3 a) Find the eigenvalues of A and list their multiplicities. b) Find a basis, Bi, for each eigenspace, E(i). c) If possible, diagonalise matrix A. (i.e find matrices P and D such that Pinv AP = D is diagonal).
For each n ∈ N, let fn : [0, 1] → [0, 1] be defined by...
For each n ∈ N, let fn : [0, 1] → [0, 1] be defined by fn(x) = 0, x > 1/n and fn(x) = 1−nx if 0 ≤ x ≤1/n. The collection {fn(x) : n ∈ N} converges to a point, call it f(x) for each x ∈ [0, 1]. Show whether {fn(x) : n ∈ N} converges to f uniformly or not.
Let A be a real n × n matrix, and suppose that every leading principal submatrix...
Let A be a real n × n matrix, and suppose that every leading principal submatrix ofA of order k < n is nonsingular. Show that A has an LU-factorisation.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT