Denseness of J= {𝐴 + 𝐵𝑇:𝐴,𝐵 ∈ 𝑀𝑘(𝑍)} in the Space of all 𝑘 × 𝑘 real matrices, where T is a fixed 𝑘 × 𝑘 matrix with irrational entries

Several countable minimal dense subsets of R are known to exist. An important example is the set Z+Zq={a+bq: a,b∈Z}, where q is an irrational number.In this paper, we establish an analogous result in the space of matrices 〖 M〗_k (R). We prove that the set J={A+BT:A,B〖∈M〗_k (Z)} is dense in M_k (R) under the metric on M_k (R) defined by 〖d(X,Y)=‖X-Y‖〗_p= (∑_(1≤i,j≤k)▒|X(i,j)-Y(i,j)|^p )^(1/p) Where T be a fixed matrix in M_k (R) whose entries are irrational numbers.