Question

Determine whether or not the set of all 2×2 matrices A with the given property is a subspace of R²*².

(a) A² =A.

Answer 1

The set of all 2x2 matrices A with the property A² = A is a subspace of ℝ²².

Step-by-step explanation:

To determine whether or not the set of all 2x2 matrices A with the property A² = A is a subspace of ℝ²², we need to check three conditions: closure under addition, closure under scalar multiplication, and the presence of the zero vector.

1. Adding two matrices: Let A and B be two matrices with the property A² = A and B² = B. Then, (A + B)² = A² + 2AB + B² = A + B + 2AB. Since A and B are in the given set, A + B and 2AB will also have the property A² = A. Therefore, the set is closed under addition.
2. Multiplying by a scalar: Let A be a matrix in the given set and k be a scalar. Then, (kA)² = k²A² = k²A. Since A is in the given set, kA will also have the property A² = A. Therefore, the set is closed under scalar multiplication.
3. Zero vector: The zero vector is the matrix of all zeros. The zero matrix also has the property A² = A since all its entries are zero. Therefore, the zero vector is present in the set.

Since the set satisfies all three conditions, it is a subspace of ℝ²².