2 questions about matrix

1. Let r be a natural number such that 2^(r 1) >= k. Then A^(2^(r 1)) = O.
We have

(I_n - A^2)(I_n A^2)(I_n A^4)...(I_n A^(2^r))
= (I_n - A^4)(I_n A^4)...(I_n A^(2^r))
= ...
= (I_n - A^(2^(r 1)))
= I_n - O = I_n.

Hence I_n - A^2 is invertible.


2. (i) Regard A as a matrix in K^(m,n). Perform Gaussian elimination to the matrix A, and let the resulting matrix be B.

Note that if we regard A as a matrix in L^(m,n) and perform Gaussian elimination to the matrix A, all the steps will be the same, and the resulting matrix is also B.

Since rank of A = number of nonzero rows in B, we conclude that the rank of A is the same when we regard it as a matirx in K^(m,n) or in L^(m,n).

(ii) Let A be the matrix with rows v_1, ... , v_m. (view them as row vectors). If these columns are linearly independent in K^n, then when A is regarded as a matrix in K^(m,n), its rank is m. If we regard v_1, ... , v_m as vectors in L^n, and the matrix A so formed a matrix in L^(m,n), then by (i), the rank of A is also equal to m. Hence v_1, ... , v_m are linearly independent when regarded as vectors in L^n.