Skip to main content

Properties of Matrix Multiplication

Matrix multiplication has several important properties that make it useful in various mathematical contexts. However, matrix multiplication differs from regular multiplication of numbers in certain aspects, such as the non-commutative property. Below are the key properties of matrix multiplication along with explanations and examples.

1. Associative Property

Matrix multiplication is associative, which means that the product of three matrices can be grouped in any way without changing the result.

Mathematically:

[ A \cdot (B \cdot C) = (A \cdot B) \cdot C ]

Where ( A ), ( B ), and ( C ) are matrices of compatible dimensions for multiplication.

Example:

Let ( A ), ( B ), and ( C ) be the following matrices:

[ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & 1 \ 4 & 2 \end{bmatrix} ]

  1. First, calculate ( B \cdot C ): [ B \cdot C = \begin{bmatrix} (2 \cdot 0 + 0 \cdot 4) & (2 \cdot 1 + 0 \cdot 2) \ (1 \cdot 0 + 3 \cdot 4) & (1 \cdot 1 + 3 \cdot 2) \end{bmatrix} = \begin{bmatrix} 0 & 2 \ 12 & 7 \end{bmatrix} ]

  2. Now, calculate ( A \cdot (B \cdot C) ): [ A \cdot \begin{bmatrix} 0 & 2 \ 12 & 7 \end{bmatrix} = \begin{bmatrix} (1 \cdot 0 + 2 \cdot 12) & (1 \cdot 2 + 2 \cdot 7) \ (3 \cdot 0 + 4 \cdot 12) & (3 \cdot 2 + 4 \cdot 7) \end{bmatrix} = \begin{bmatrix} 24 & 16 \ 48 & 34 \end{bmatrix} ]

  3. Alternatively, calculate ( (A \cdot B) \cdot C ): [ A \cdot B = \begin{bmatrix} (1 \cdot 2 + 2 \cdot 1) & (1 \cdot 0 + 2 \cdot 3) \ (3 \cdot 2 + 4 \cdot 1) & (3 \cdot 0 + 4 \cdot 3) \end{bmatrix} = \begin{bmatrix} 4 & 6 \ 10 & 12 \end{bmatrix} ]

Now, multiply ( (A \cdot B) \cdot C ): [ \begin{bmatrix} 4 & 6 \ 10 & 12 \end{bmatrix} \cdot \begin{bmatrix} 0 & 1 \ 4 & 2 \end{bmatrix} = \begin{bmatrix} (4 \cdot 0 + 6 \cdot 4) & (4 \cdot 1 + 6 \cdot 2) \ (10 \cdot 0 + 12 \cdot 4) & (10 \cdot 1 + 12 \cdot 2) \end{bmatrix} = \begin{bmatrix} 24 & 16 \ 48 & 34 \end{bmatrix} ]

Since ( A \cdot (B \cdot C) = (A \cdot B) \cdot C ), the associative property is verified.

2. Distributive Property

Matrix multiplication is distributive over matrix addition. This means that multiplying a matrix by the sum of two matrices is the same as multiplying the matrix by each matrix individually and then adding the results.

Mathematically:

[ A \cdot (B + C) = A \cdot B + A \cdot C ]

Where ( A ), ( B ), and ( C ) are matrices of compatible dimensions.

Example:

Let ( A ), ( B ), and ( C ) be the following matrices:

[ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 \ 0 & 3 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \ 4 & 2 \end{bmatrix} ]

  1. First, calculate ( B + C ): [ B + C = \begin{bmatrix} 2+1 & 1+0 \ 0+4 & 3+2 \end{bmatrix} = \begin{bmatrix} 3 & 1 \ 4 & 5 \end{bmatrix} ]

  2. Now, calculate ( A \cdot (B + C) ): [ A \cdot \begin{bmatrix} 3 & 1 \ 4 & 5 \end{bmatrix} = \begin{bmatrix} (1 \cdot 3 + 2 \cdot 4) & (1 \cdot 1 + 2 \cdot 5) \ (3 \cdot 3 + 4 \cdot 4) & (3 \cdot 1 + 4 \cdot 5) \end{bmatrix} = \begin{bmatrix} 11 & 11 \ 25 & 23 \end{bmatrix} ]

  3. Now, calculate ( A \cdot B + A \cdot C ): [ A \cdot B = \begin{bmatrix} (1 \cdot 2 + 2 \cdot 0) & (1 \cdot 1 + 2 \cdot 3) \ (3 \cdot 2 + 4 \cdot 0) & (3 \cdot 1 + 4 \cdot 3) \end{bmatrix} = \begin{bmatrix} 2 & 7 \ 6 & 15 \end{bmatrix} ]

[ A \cdot C = \begin{bmatrix} (1 \cdot 1 + 2 \cdot 4) & (1 \cdot 0 + 2 \cdot 2) \ (3 \cdot 1 + 4 \cdot 4) & (3 \cdot 0 + 4 \cdot 2) \end{bmatrix} = \begin{bmatrix} 9 & 4 \ 19 & 8 \end{bmatrix} ]

Now, sum the two products: [ A \cdot B + A \cdot C = \begin{bmatrix} 2 & 7 \ 6 & 15 \end{bmatrix} + \begin{bmatrix} 9 & 4 \ 19 & 8 \end{bmatrix} = \begin{bmatrix} 11 & 11 \ 25 & 23 \end{bmatrix} ]

Since ( A \cdot (B + C) = A \cdot B + A \cdot C ), the distributive property is verified.

3. Non-Commutative Property

Unlike regular multiplication, matrix multiplication is not commutative. This means that, in general, ( A \cdot B \neq B \cdot A ).

Mathematically:

[ A \cdot B \neq B \cdot A ]

Example:

Let ( A ) and ( B ) be the following matrices:

[ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \ 2 & 3 \end{bmatrix} ]

  1. Calculate ( A \cdot B ): [ A \cdot B = \begin{bmatrix} (1 \cdot 0 + 2 \cdot 2) & (1 \cdot 1 + 2 \cdot 3) \ (3 \cdot 0 + 4 \cdot 2) & (3 \cdot 1 + 4 \cdot 3) \end{bmatrix} = \begin{bmatrix} 4 & 7 \ 8 & 15 \end{bmatrix} ]

  2. Now, calculate ( B \cdot A ): [ B \cdot A = \begin{bmatrix} (0 \cdot 1 + 1 \cdot 3) & (0 \cdot 2 + 1 \cdot 4) \ (2 \cdot 1 + 3 \cdot 3) & (2 \cdot 2 + 3 \cdot 4) \end{bmatrix} = \begin{bmatrix} 3 & 4 \ 11 & 16 \end{bmatrix} ]

Clearly, ( A \cdot B \neq B \cdot A ), which verifies the non-commutative property of matrix multiplication.

Conclusion

  • Associative Property: The order in which matrices are grouped for multiplication does not change the result.
  • Distributive Property: Matrix multiplication distributes over addition.
  • Non-Commutative Property: The order of multiplication matters; in general, ( A \cdot B \neq B \cdot A ).

These properties are fundamental in matrix operations and are widely used in linear algebra, computer graphics, and many applications in science and engineering.

No Comments
Back to top