Understanding Matrix Multiplication: Commutative, Associative, and Distributive Properties

Is matrix multiplication for square matrices commutative, associative, or distributive?

Matrix multiplication is neither commutative nor associative

Matrix multiplication is neither commutative nor associative. However, it does follow the distributive property.

1. Commutative Property:
Matrix multiplication is not commutative. This means that the order in which you multiply matrices matters. In general, if you have matrices A and B, the product AB is not necessarily equal to the product BA. In other words, if AB is defined, BA may not be defined or may produce a different result.

For example, let’s consider two square matrices A and B:
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]

If we calculate AB, we get:
AB = [[1*5 + 2*7, 1*6 + 2*8], [3*5 + 4*7, 3*6 + 4*8]]
= [[19, 22], [43, 50]]

Now, let’s calculate BA:
BA = [[5*1 + 6*3, 5*2 + 6*4], [7*1 + 8*3, 7*2 + 8*4]]
= [[23, 34], [31, 46]]

As you can see, AB and BA are not equal, indicating that matrix multiplication is not commutative.

2. Associative Property:
Matrix multiplication is not associative either. This means that the order in which you group and multiply matrices matters. In general, for matrices A, B, and C, (AB)C is not equal to A(BC). The order of multiplications will lead to different results.

For example, consider three square matrices A, B, and C:
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
C = [[9, 10], [11, 12]]

If we calculate (AB)C, we get:
(AB)C = ([[19, 22], [43, 50]])C
= [[19*9 + 22*11, 19*10 + 22*12], [43*9 + 50*11, 43*10 + 50*12]]
= [[587, 634], [1295, 1402]]

Now, let’s calculate A(BC):
A(BC) = A([[95, 106], [127, 142]])
= [[1*95 + 2*127, 1*106 + 2*142], [3*95 + 4*127, 3*106 + 4*142]]
= [[319, 450], [685, 966]]

As you can see, (AB)C and A(BC) are not equal, indicating that matrix multiplication is not associative.

3. Distributive Property:
Matrix multiplication follows the distributive property. This property states that for matrices A, B, and C, the product A(B + C) is equal to AB + AC.

For example, let’s consider three square matrices A, B, and C:
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
C = [[9, 10], [11, 12]]

If we calculate AB + AC, we get:
AB = [[1*5 + 2*7, 1*6 + 2*8], [3*5 + 4*7, 3*6 + 4*8]]
= [[19, 22], [43, 50]]

AC = [[1*9 + 2*11, 1*10 + 2*12], [3*9 + 4*11, 3*10 + 4*12]]
= [[31, 34], [69, 76]]

AB + AC = [[19, 22], [43, 50]] + [[31, 34], [69, 76]]
= [[19 + 31, 22 + 34], [43 + 69, 50 + 76]]
= [[50, 56], [112, 126]]

Now, let’s calculate A(B + C):
B + C = [[5 + 9, 6 + 10], [7 + 11, 8 + 12]]
= [[14, 16], [18, 20]]

A(B + C) = [[1*14 + 2*18, 1*16 + 2*20], [3*14 + 4*18, 3*16 + 4*20]]
= [[50, 56], [112, 126]]

As you can see, AB + AC and A(B + C) are equal, indicating that matrix multiplication follows the distributive property.

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