In this article, we will explain what matrix multiplication is all about and use tips, tricks and example exercises to show you how you are guaranteed to get the right result. This article belongs to the subject Mathematics and expands on the subject of matrices.

## What is a matrix anyway?

Data can be presented in a structured way with the help of matrices. Data is usually prepared in tabular form, although this information can also be presented in simplified form in the form of a matrix. A matrix is composed of **m lines** and **n columns** and **their elements**. One **(mxn) matrix **has the following form:

matrix representation:

The elements of a matrix are defined by real numbers or their placeholders. When specifying or writing a matrix, please note that **first the line number **and **then the column number** is specified. For example, the element is found in the third row and fifth column of an (mxn) matrix.

A column vector is a matrix with **only one column** and accordingly under a row vector a matrix with **just one line **.

Matrix notation, or the matrix itself, is a short, concise, and pleasant notation for mathematical problems, methods, and data. Matrices can be added, subtracted, and multiplied, however **not divided **will.

## When do I need matrix multiplication?

For example, linear systems of equations or linear mappings can be represented in a simplified manner in matrix form through structural data processing. This allows various arithmetic operations to be carried out with matrices. Elementary arithmetic operations such as matrix addition, matrix subtraction or matrix multiplication can be carried out under certain conditions.

Matrix addition and matrix subtraction are only possible if the **Number of rows and columns** of the two matrices together **to match**. For the multiplication of two matrices A and B the **Number of columns of matrix A **with the **Number of rows of matrix B** agree or vice versa. It should be noted that a division of two or more matrices from each other is not defined in mathematics!

In connection with the terms «multiplication» and «matrix» it should be noted that a matrix A can be multiplied in different ways.

One distinguishes:

- The multiplication of a matrix by a scalar (a number)

- The multiplication of a matrix by a vector

- The multiplication of a matrix by a matrix

The method of matrix multiplication, on the other hand, describes the multiplicative linking of matrices and is used, for example, in the solution of linear systems of equations or in the concatenation of linear mappings.

If two matrices A and B are multiplied with each other, the resulting matrix C is created. This matrix describes the matrix product of A and B. You can now find out exactly how the arithmetic operation behind matrix multiplication works.

## What should I watch out for when doing matrix multiplication?

Matrices can only be multiplied with each other if the **number of columns in matrix A matches the number of rows in matrix B** or the other way around. The matrix A is multiplied by each column vector of the matrix B in turn. In order to clearly describe the operation of matrix multiplication, you can see below the procedure for multiplying a (3×2) matrix with a square (2×2) matrix.

scheme:

Given is the matrix and the matrix .

- Column Vector:
- Column Vector:

The resulting matrix C, the matrix product, has the same number of rows as matrix A and the same number of columns as matrix B.

It should also be noted that the communicative law is not valid for matrix multiplication, ie

The following arithmetic laws are valid in connection with matrix multiplication:

- for the associated
**unit matrices**E - for the
**zero matrix**O - for the
**associative law** - for the
**distributive law** - for
**scalars**s∈R

In the following we now see the procedure for matrix multiplication using an example based on the Falk scheme.

## Matrix multiplication – application and exercise

Since in school mathematics, when multiplying a matrix, for a better understanding one usually** Scheme according to Falk** applies, now let’s see an example. Falk’s scheme is described in a table that provides visual aid in the calculation. In the following, a (3×2) matrix is multiplied by a square (2×2) matrix. We start step by step with the calculation method and a functional example.

### Matrix multiplication – functional example according to Falk’s scheme

In order to be able to understand matrix multiplication optimally, we are now presenting a recipe, Falk’s scheme, for the calculation using a functional example:

**Calculation method and functional example:**

Given is the matrix and the matrix .

**Step1:**Write down the matrices to be multiplied, offset in height and next to each other according to Falk’s scheme.**Step2:**Multiply the first row of the first matrix elementary by the first column of the second matrix 1st row of columns- Step3: Carry out this scheme with all relevant rows (m-rows) of the first matrix and the first column of the second matrix.
- Step4: Multiply the first row of the first matrix elementary by the second column of the second matrix -> 2nd row of columns
- Step5: Carry out this scheme with all relevant rows (m-rows) of the first matrix and the second column of the second matrix.
- Step6: Multiply the first row of the first matrix elementary by all relevant columns of the second matrix up to the last column (n-column)
- Step7: Carry out this scheme with all relevant rows (m-rows) of the first matrix and with all relevant columns (n-columns) of the second matrix.

FINISHED!

Congratulations, you did the hardest part. You now know how matrix multiplication works.

Of course, matrix multiplication is like any other mathematical method. You gotta practice, practice, practice…

You can find exercises and helpful literature on the subject of matrix multiplication here. (Link: https:///…)

## our recommendation

It is incredibly important that you master the basic arithmetic operations related to matrices. You must be able to easily add and subtract matrices together. If you still feel unsure about multiplying matrices, first familiarize yourself with s-multiplication and the matrix-vector product.

## summary

In matrix multiplication, two or more matrices are multiplied at the same time. To multiply two matrices A and B, the number of columns in matrix A must match the number of rows in matrix B, or vice versa. In school mathematics, the calculation is usually carried out in a simplified manner according to Falk’s scheme.

If you have any questions, please use our comment section! You can find more exercises in the free* content of STARK and in our index cards on this and many other topics!

## INSIDER TIP

“ Hey, cool that you are interested in the topic of matrix multiplication! Did you know that two matrices can be added, multiplied, and subtracted, but not divided? The division of matrices is not mathematically defined. You can find more information about this on this Learning Page! If you have any questions, please use our comment section! Check it out! ”

Leon Jerg

Study Smarter Institute