Question

Find two new matrices m and n that have the so-called commutative property mn = nm that was noticed in part (c) of the preceding.

a) Matrix m = [1 0; 0 1], Matrix n = [1 0; 0 1]
b) Matrix m = [1 2; 3 4], Matrix n = [0 1; 1 0]
c) Matrix m = [1 2; 2 3], Matrix n = [3 1; 2 0]
d) Matrix m = [0 1; 1 0], Matrix n = [1 2; 3 4]

Answer 1

The matrices that satisfy the commutative property are Matrix m = [1 0; 0 1] and Matrix n = [1 0; 0 1].

Step-by-step explanation:

The commutative property states that the order of operations does not affect the result. In the case of matrices, the commutative property means that the order of multiplication does not affect the product. To find two matrices that satisfy the commutative property, we can check the given options:

1. Matrix m = [1 0; 0 1] and Matrix n = [1 0; 0 1]: When we multiply these two matrices, mn = [1 0; 0 1] and nm = [1 0; 0 1]. They are equal, so they satisfy the commutative property.
2. Matrix m = [1 2; 3 4] and Matrix n = [0 1; 1 0]: When we multiply these two matrices, mn = [2 1; 4 3] and nm = [2 1; 4 3]. They are not equal, so they do not satisfy the commutative property.
3. Matrix m = [1 2; 2 3] and Matrix n = [3 1; 2 0]: When we multiply these two matrices, mn = [7 1; 15 4] and nm = [5 8; 6 10]. They are not equal, so they do not satisfy the commutative property.
4. Matrix m = [0 1; 1 0] and Matrix n = [1 2; 3 4]: When we multiply these two matrices, mn = [3 4; 1 2] and nm = [3 1; 4 2]. They are not equal, so they do not satisfy the commutative property.

Therefore, the matrices that satisfy the commutative property are Matrix m = [1 0; 0 1] and Matrix n = [1 0; 0 1].