Question

1) Suppose A and B are 2x2 matrices. Show that the set {X: X AB=B AX} is a subspace of the vector space of 2x2 matrices.

2) Define addition on R² as follows: (x1,x2)+(y1,y2)= (x1+y2, x2+y1). Define scalar multiplication in the usual way. Is R² a vector space under these operations?

Answer 1

To show that the set {X: X AB = B AX} is a subspace of the vector space of 2x2 matrices, we need to show that it contains the zero vector and is closed under addition and scalar multiplication.

Step-by-step explanation:

To show that the set {X: X AB = B AX} is a subspace of the vector space of 2x2 matrices, we need to show that it satisfies three conditions:

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

To determine if the set contains the zero vector, we set X = 0 and check if it satisfies the equation AB = BAX. Since the zero matrix multiplied by any matrix results in the zero matrix, this equation holds.

To show closure under addition, we take two matrices X1 and X2 from the set and check if their sum X1 + X2 satisfies the equation (X1 + X2)AB = B(X1 + X2)A. By distributing the matrix operations, the equation simplifies to X1AB + X2AB = BX1A + BX2A. Since X1AB = BX1A and X2AB = BX2A (because X1 and X2 are in the set), the equation holds.

To show closure under scalar multiplication, we take a matrix X from the set and a scalar c, and check if the scalar multiple cX satisfies the equation (cX)AB = B(cX)A. By distributing the matrix operations, the equation simplifies to c(XAB) = B(cXA). Since XAB = BXA (because X is in the set), the equation holds.