# Category:Conventional Matrix Multiplication

This category contains results about **(conventional) matrix multiplication**.

Let $\struct {R, +, \circ}$ be a ring.

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$.

Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$.

Then the **matrix product of $\mathbf A$ and $\mathbf B$** is written $\mathbf A \mathbf B$ and is defined as follows.

Let $\mathbf A \mathbf B = \mathbf C = \sqbrk c_{m p}$.

Then:

- $\ds \forall i \in \closedint 1 m, j \in \closedint 1 p: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$

Thus $\sqbrk c_{m p}$ is the $m \times p$ matrix where each entry $c_{i j}$ is built by forming the (ring) product of each entry in the $i$'th row of $\mathbf A$ with the corresponding entry in the $j$'th column of $\mathbf B$ and adding up all those products.

This operation is called **matrix multiplication**, and $\mathbf C$ is the **matrix product** of $\mathbf A$ with $\mathbf B$.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Conventional Matrix Multiplication"

The following 22 pages are in this category, out of 22 total.

### E

### M

- Matrix Multiplication Distributes over Matrix Addition
- Matrix Multiplication is Associative
- Matrix Multiplication is Homogeneous of Degree 1
- Matrix Multiplication is not Commutative
- Matrix Multiplication on Square Matrices over Trivial Ring is Commutative
- Matrix Multiplication over Order n Square Matrices is Closed
- Matrix Product with Adjugate Matrix