CALCULATE

CALCULATE

** 3x3 matrix multiplication calculator** uses two matrices $A$ and $B$ and calculates the product $AB$. It is an online math tool specially programmed to perform multiplication operation between the two matrices $A$ and $B$. The matrix multiplication is not commutative operation. In the matrix multiplication $AB$, the number of columns in matrix $A$ must be equal to the number of rows in matrix $B$.

It is necessary to follow the next steps:

- Enter two matrices in the box. Elements of matrices must be real numbers.
- Press the "
**GENERATE WORK**" button to make the computation; - 3x3 matrix multiplication calculator will give the product of the first and second entered matrix.

The product of two matrices $A=(a_{ij})_{3\times 3}$ and $B=(a_{ij})_{3\times 3}$ is determined by the following formula
$$\begin{align}&\left(
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right)\cdot
\left(
\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} &b_{32} & b_{33} \\
\end{array}
\right)\\&= \left(\begin{array}{ccc}
a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31}& a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}& a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33} \\
a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31} &a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32}& a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}\\
a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31} &a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32} & a_{31}b_{13}+a_{32}b_{23}+a_{33}b_{33}\\
\end{array}\right)\end{align}$$

Matrices are a powerful tool in mathematics, science and life. Matrices are everywhere and they have significant applications. For example, spreadsheet such as Excel or written a table represents a matrix. The word "matrix" is the Latin word and it means "womb". This term was introduced by J. J. Sylvester (English mathematician) in 1850.
The first need for matrices was in the studying of systems of simultaneous linear equations.

A matrix is a rectangular array of numbers, arranged in the following way
$$A=\left(
\begin{array}{cccc}
a_{11} & a_{12} & \ldots&a_{1n} \\
a_{21} & a_{22} & \ldots& a_{2n} \\
\ldots &\ldots &\ldots&\ldots\\
a_{m1} & a_{m2} & \ldots&a_{mn} \\
\end{array}
\right)=\left[
\begin{array}{cccc}
a_{11} & a_{12} & \ldots&a_{1n} \\
a_{21} & a_{22} & \ldots& a_{2n} \\
\ldots &\ldots &\ldots&\ldots\\
a_{m1} & a_{m2} & \ldots&a_{mn} \\
\end{array}
\right]$$
There are two notation of matrix: in parentheses or box brackets. The terms in the matrix are called its entries or its elements.
Matrices are most often denoted by upper-case letters, while the corresponding lower-case letters, with two subscript indices, are the elements of matrices.
For examples, matrices are denoted by $A,B,\ldots Z$ and its elements by $a_{11}$ or $a_{1,1}$, etc. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.

The size of a matrix is a Descartes product of the number of rows and columns that it contains.
A matrix with $m$ rows and $n$ columns is called an $m\times n$ matrix. In this case $m$ and $n$ are its dimensions.
If a matrix consists of only one row, it is called a row matrix. If a matrix consists
only one column is called a column matrix. A matrix
which contains only zeros as elements is called a zero matrix.
A square matrix is a matrix with the same number of rows and columns. A square matrix with all elements as zeros except for the main diagonal, which has only ones, is called an identity matrix. For instance, the following matrices $$I_1=(1),\; I_2=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}
\right),\ldots ,I_n=\left(
\begin{array}{cccc}
1 & 0 & \ldots & 0 \\
0 & 1 & \ldots & 0 \\
\ldots & \ldots & \ldots & \ldots \\
0 & 0 & \ldots & 1 \\
\end{array}
\right)$$
are identity matrices of size $1\times1$, $2\times 2, \ldots$ $n\times n$, respectively.

Many operations with matrices make sense only if the matrices have suitable dimensions. In other words, they should be the same size, with the same number of rows and the same number of columns.

When we deal with matrix multiplication, matrices $A=(a_{ij})_{m\times p}$ with $m$ rows, $p$ columns and $B=(b_{ij})_{r\times n}$ with $r$ rows, $n$ columns can be multiplied if and only if $p=r$. This means, that the number of columns of the first matrix, $A$, must be equal to the number of rows of the second matrix, $B$.
The product of these matrix is a new matrix that has the same number of rows as the first matrix, $A$, and the same number of columns as the second matrix, $B$.
So, the corresponding product $C=A\cdot B$ is a matrix of size $m\times n$. Elements $c_{ij}$ of this matrix are
$$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}\ldots+a_{ip}b_{pj}\quad\mbox{for}\;i=1,\ldots,m,\;j=1,\ldots,n.$$
For example, $3\times 3$ matrix multiplication is determined by the following formula
$$\begin{align}&\left(
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right)\cdot
\left(
\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} &b_{32} & b_{33} \\
\end{array}
\right)\\&= \left(\begin{array}{ccc}
a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31}& a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32}& a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33} \\
a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31} &a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32}& a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33}\\
a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31} &a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32} & a_{31}b_{13}+a_{32}b_{23}+a_{33}b_{33}\\
\end{array}\right)\end{align}$$

**Properties of Matrix Multiplication**

- Matrix multiplication is not commutative in general, $AB \not BA$. In some cases, it is possible that the product $AB$ exists, while the product $BA$ does not exist. Both products $AB$ and $BA$ are defined if and only if the matrices $A$ and $B$ are square matrices of a same size.
- If $A=(a_{ij})_{mn}$, $B=(b_{ij})_{np}$ and $C=(c_{ij})_{pk}$, then matrix multiplication is associative, i.e. $$A(BC)=(AB)C$$
- If $A=(a_{ij})_{mn}$, $B=(b_{ij})_{np}$, $C=(c_{ij})_{np}$ and $D=(d_{ij})_{pq}$, then the matrix multiplication is distributive with respect of matrix addition, i.e. $$\begin{align} A(B+C)&=AB+AC\\ (B+C)D&=BD+CD\end{align}$$
- If $A_{n\times n}$ is a square matrix, it exists an identity matrix $I_{n\times n}$ such that $$AI=IA=A$$

The matrix multiplication work with steps shows the complete step-by-step calculation for finding the product $AB$ of two $3\times 3$ matrices $A$ and $B$ using the matrix multiplication formula. For any other matrices, just supply elements of $2$ matrices whose elements are real numbers and click on the GENERATE WORK button. The grade school students and people who study math use this matrix multiplication calculator to generate the work, verify the results of multiplication matrices derived by hand, or do their homework problems efficiently. The grade school students can also use this calculator for solving linear equations.

One of the main application of matrix multiplication is in solving systems of linear equations. Transformations in two or three dimensional Euclidean geometry can be represented by $2\times 2$ or $3\times 3$ matrices. Dilation, translation, axes reflections, reflection across the $x$-axis, reflection across the $y$-axis, reflection across the line $y=x$, rotation, rotation of $90^o$ counterclockwise around the origin, rotation of $180^o$ counterclockwise around the origin, etc, use $2\times 2$ and $3\times 3$ matrix multiplications.

**Practice Problem 1 : **

Find the product $AB$ for $$A=\left(
\begin{array}{cc}
4& 20 \\
5 & 5 \\
2 &-6 \\
\end{array}
\right)\quad\mbox{and}\quad B=\left(
\begin{array}{cc}
3 & 2 \\
3 & 3 \\
\end{array}
\right)$$
**Practice Problem 2 : **

Find the image of a transformation of the vertex matrix $\left(
\begin{array}{cc}
3 & 2 \\
3 & 3 \\
\end{array}
\right)$ when it is rotated $90^o$ counterclockwise around the origin.

The matrix multiplication calculator, formula, example calculation (work with steps), real world problems and practice problems would be very useful
for grade school students (K-12 education) to understand the matrix multiplication of two or more matrices. Using this concept they can solve systems of linear equations and other linear algebra problems in physics, engineering and computer science.