Properties of Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra with applications in various fields such as physics, engineering, computer science, and economics. Understanding the properties of matrix multiplication is crucial for solving linear equations, transforming geometric objects, and performing complex calculations in scientific computing.

Basic Definition

Given two matrices $A$ and $B$, where $A$ is of size $m \times n$ and $B$ is of size $n \times p$, the product $AB$ is defined as:

$$ (AB){ij} = \sum{k=1}^{n} A_{ik}B_{kj} $$

This means that the element in the $i$-th row and $j$-th column of the matrix product $AB$ is the sum of the products of corresponding elements from the $i$-th row of $A$ and the $j$-th column of $B$.

Properties of Matrix Multiplication

Matrix multiplication is not as straightforward as scalar multiplication or addition of matrices. It has unique properties that are important to understand:

Property	Description	Mathematical Representation
Associativity	The product of three or more matrices is associative.	$(AB)C = A(BC)$
Distributivity	Matrix multiplication is distributive over addition.	$A(B + C) = AB + AC$ and $(A + B)C = AC + BC$
Non-commutativity	In general, matrix multiplication is not commutative.	$AB \neq BA$
Identity Element	There exists an identity matrix for which the product with any matrix leaves it unchanged.	$AI = IA = A$
Dimension Compatibility	The number of columns in the first matrix must be equal to the number of rows in the second matrix.	If $A$ is $m \times n$, then $B$ must be $n \times p$ for $AB$ to be defined.
Zero Matrix	Multiplying any matrix by a zero matrix results in a zero matrix.	$A0 = 0$ and $0B = 0$
Transpose	The transpose of a matrix product is the product of the transposes in reverse order.	$(AB)^T = B^T A^T$
Inverse	If two square matrices have inverses, the inverse of their product is the product of their inverses in reverse order.	$(AB)^{-1} = B^{-1} A^{-1}$

Examples

Associativity

Given matrices $A$, $B$, and $C$, where the product $ABC$ is defined, we have:

$$ (AB)C = A(BC) $$

Distributivity

For matrices $A$, $B$, and $C$ of compatible dimensions, we can distribute the multiplication over addition as follows:

$$ A(B + C) = AB + AC $$

Non-commutativity

For matrices $A$ and $B$ of size $2 \times 2$:

$$ A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix} $$

Their products are:

$$ AB = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix} $$

$$ BA = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix} \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} = \begin{pmatrix} 23 & 34 \ 31 & 46 \end{pmatrix} $$

Clearly, $AB \neq BA$.

Identity Element

For any matrix $A$ of size $2 \times 2$:

$$ A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \quad I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $$

The product with the identity matrix is:

$$ AI = IA = A $$

Dimension Compatibility

If $A$ is a $2 \times 3$ matrix and $B$ is a $3 \times 4$ matrix, then $AB$ is defined and results in a $2 \times 4$ matrix. However, $BA$ is not defined since the number of columns in $B$ does not match the number of rows in $A$.

Zero Matrix

For any matrix $A$ of size $2 \times 2$:

$$ A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \quad 0 = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} $$

The product with a zero matrix is:

$$ A0 = 0A = 0 $$

Transpose

For matrices $A$ and $B$:

$$ (AB)^T = B^T A^T $$

Inverse

For invertible matrices $A$ and $B$:

$$ (AB)^{-1} = B^{-1} A^{-1} $$

Understanding these properties is essential for manipulating matrices and solving complex problems in various mathematical and practical contexts. It is important to remember that these properties may not hold for operations other than matrix multiplication, and care must be taken when performing calculations involving matrices.