Matrix multiplication is an Associative property.

Matrix multiplication is a fundamental operation in linear algebra, and it follows certain properties. One of these properties is associativity, which means that the order in which matrix multiplication is performed does not affect the result. In other words, when multiplying three matrices, the grouping of the operations does not matter.

Associativity of matrix multiplication:

Let's consider three matrices A, B, and C. The product of these matrices can be written as (AB)C or A(BC). According to the associative property, both of these expressions should give the same result.

Mathematically, if A is an m × n matrix, B is an n × p matrix, and C is a p × q matrix, then:

(AB)C = A(BC)

Proof:

To prove the associative property of matrix multiplication, we need to show that both (AB)C and A(BC) yield the same result.

Let's consider the product (AB)C:

(AB)C = D

Here, D is an m × q matrix.

Now, let's consider the product A(BC):

A(BC) = E

Here, E is also an m × q matrix.

To prove the associative property, we need to show that D and E are equal.

To do this, we can expand the expressions (AB)C and A(BC) using the properties of matrix multiplication:

(AB)C = A(BC)
(ABC) = (ABC)

Since both sides of the equation are equal, we can conclude that matrix multiplication is an associative property.

Conclusion:

Matrix multiplication is an associative property, which means that the order of multiplication does not affect the result. This property is important in various applications of linear algebra, such as solving systems of linear equations, finding eigenvalues and eigenvectors, and performing transformations in computer graphics.