Question

Determine whether the subset of M(n, n) is a subspace of M(n, n) with the standard operations of matrix addition and scalar multiplication. The set of all n x n matrices with integer entries.

Answer 1

Yes, the subset of M(n, n) consisting of all n x n matrices with integer entries is a subspace of M(n, n) with the standard operations of matrix addition and scalar multiplication. This is proven by showing that the subset is closed under addition, closed under scalar multiplication, and contains the zero vector.

Step-by-step explanation:

Yes, the subset of M(n, n) consisting of all n x n matrices with integer entries is a subspace of M(n, n) with the standard operations of matrix addition and scalar multiplication.

To prove this, we need to show that the subset satisfies three conditions:

1. The subset is closed under addition.
2. The subset is closed under scalar multiplication.
3. The subset contains the zero vector.

First, let A and B be matrices in the subset. The sum of A and B, denoted as A + B, is also an n x n matrix with integer entries. Therefore, the subset is closed under addition.

Second, let A be a matrix in the subset and c be a scalar. The product of A and c, denoted as cA, is also an n x n matrix with integer entries. Therefore, the subset is closed under scalar multiplication.

Finally, the zero matrix, which consists of all zeros, is an n x n matrix with integer entries. Therefore, the subset contains the zero vector.

Since the subset satisfies all three conditions, it is a subspace of M(n, n) with the standard operations of matrix addition and scalar multiplication.