Linear Algebra!

[little_bit_crazy]

1. Show that the elements sent to zero by a linear transformation form a linear subspace.

2. Let A be a square matrix. By A^n, we mean A multipled by a itself n time. By A^0, we mean Id. Show that if a matrix A satisfies A^n = Id for some n greater than or equal to 1, then A is invertible.

3. Show that if a matrix A^n satifies A^n = 0 for some n, than A is not invertible.

if you could help me with 1, 2, or 3 that would be amazing!!

Thanks

 

[Matt]

For the second problem:

2. Let A be a square matrix. By A^n, we mean A multipled by a itself n time. By A^0, we mean Id. Show that if a matrix A satisfies A^n = Id for some n greater than or equal to 1, then A is invertible.

Let B = A^n-1. Then AB = BA = I. Thus A is invertible.

 

[Matt]

For the third problem, you can do a proof by contradiction.

3. Show that if a matrix A^n satifies A^n = 0 for some n, than A is not invertible.

Suppose to the contrary that A is in fact invertible, and let B be the inverse of A. Then what is BⁿAⁿ?