Matrix multiplication involves multiplying two matrices (arrays of numbers) together to produce a third matrix. It is widely used in mathematics, computer programming, and engineering. The question of whether matrix multiplication is commutative (able to be done in either order) has been an active research topic for many years.
Definition of Matrix Multiplication
Matrix multiplication is the process of multiplying two or more matrices. It is a form of linear algebra and has applications in many areas. The definition of matrix multiplication is as follows: when two matrices A and B with appropriate size are multiplied, the result is a matrix C whose elements are obtained by the summation of the products of the corresponding element in each row of A and column of B. The following equation demonstrates the general formula for matrix multiplication:
$$C_{ij} = \sum_{k=1}^{m}A_{ik}B_{kj}$$
Here $A$ has $n$ rows and $m$ columns, and $B$ has $m$ rows and $k$ columns, then $C$ will have size $n \times k$.
Matrix multiplication is an important operation in linear algebra and is used in many areas such as engineering, physics, and computer science. It is also used in machine learning algorithms such as neural networks and deep learning. Matrix multiplication is a powerful tool for solving linear equations and can be used to solve systems of linear equations.
Conditions for Commutativity
In order for matrix multiplication to be commutative, the number of columns in the first matrix must equal the number of rows in the second matrix. This means that if matrix A has size $m \times n$ and matrix B has size $n \times p$, then matrix multiplication can be done in either order. For example, AB = BA.
Examples of Matrix Multiplication
To get a better understanding of matrix multiplication and its associated rules, let’s take a look at some examples. Consider the following two matrices:
$$A = \begin{bmatrix}1 & 3\\-1 & 2\end{bmatrix} \text{ and } B = \begin{bmatrix}1 & 0\\4 & 5\end{bmatrix} $$
In this case, matrix A has size $2 \times 2$ and matrix B has size $2 \times 2$. This means we can multiply them in either order, AB or BA. If we do AB first, we get the following result:
$$AB = \begin{bmatrix}1 & 3\\-1 & 2\end{bmatrix} \begin{bmatrix}1 & 0\\4 & 5\end{bmatrix} = \begin{bmatrix}13 & 15\\2 & 1\end{bmatrix}$$
If we do BA, we get the following result:
$$BA = \begin{bmatrix}1 & 0\\4 & 5\end{bmatrix} \begin{bmatrix}1 & 3\\-1 & 2\end{bmatrix} = \begin{bmatrix}-5 & 11\\19 & 23\end{bmatrix}$$
We can see from these examples that matrix multiplication is commutative, because both orders give the same result. This result holds true in general – when the number of columns in the first matrix equals the number of rows in the second matrix.
Properties of Commutative Matrix Multiplication
When two matrices can be multiplied together in either order, they have certain properties that are inherent to this situation. The commutative property states that if A and B are two matrices such that AB = BA, then A and B are said to commute with each other. In addition, the associative property states that if A, B and C are three matrices such that (AB)C = A(BC), then A, B and C are said to be associative with each other.
These properties are useful because they allow us to simplify some expressions involving more than two matrices. For example, if A, B, C and D are four matrices such that (AB)(CD) = (AC)(BD), then we can simplify this expression to A(BCD). This is due to the associativity property.
Applications of Matrix Multiplication
Matrix multiplication is used in many different fields. In computer science and engineering, it is used to facilitate efficient data processing, such as image processing, facial recognition, and artificial intelligence. In mathematics, it is used to solve linear equations, calculate eigenvalues and eigenvectors, and study properties of geometric shapes. In physics, it is used in quantum mechanics and other areas.
Alternatives to Commutative Matrix Multiplication
If two matrices don’t commute with each other, then there are other ways to multiply them. One option is to use a technique called Gaussian Elimination, which involves transforming one or both of the matrices into an upper triangular form before multiplying them. Another option is to use a different type of multiplication algorithm that is not commutative.
Summary
In conclusion, matrix multiplication is commutative when the number of columns in the first matrix matches the number of rows in the second matrix. This means that if matrix A has size $m \times n$ and matrix B has size $n \times p$, then matrix multiplication can be done in either order (AB = BA). Other conditions for commutativity include having both matrices be square and having all elements be real numbers. There are other alternatives available if two matrices don’t commute with each other, including Gaussian Elimination and other algorithms.
