CALCULATE

CALCULATE

** 3x3 matrix addition calculator** uses two $3\times 3$ matrices and calculates their sum. It is an online math tool specially
programmed to perform matrix addition between the two $3\times 3$ matrices.

It is necessary to follow the next steps:

- Enter two $3\times3$ matrices in the box. Elements of matrices must be real numbers.
- Press the "
**GENERATE WORK**" button to make the computation; - 3x3 matrix addition calculator will give the sum of two $3\times3$ matrices represented as matrix.

The sum of two matrices $A=[a_{ij}]_{3\times3}$ and $B=[a_{ij}]_{3\times3}$ is determined by the following formula
$$\begin{align}&\left(
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right)+
\left(
\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} &b_{32} & b_{33} \\
\end{array}
\right)= \left(\begin{array}{ccc}
a_{11}+b_{11}& a_{12}+b_{12}& a_{13}+b_{13} \\
a_{21}+b_{21} &a_{22}+b_{22}& a_{23}+b_{23}\\
a_{31}+b_{31} &a_{32}+b_{32} & a_{33}+b_{33}\\
\end{array}\right)\end{align}$$

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.

Matrices $A$ and $B$ can be added if and only if their sizes are equal. Their sum is a matrix $C=A+B$ with elements
$$c_{ij}=a_{ij}+b_{ij}$$
The matrix of sum has the same size as the matrices $A$ and $B$.
This means, each element in $C$ is equal to the sum of the elements in $A$ and $B$ that are located in corresponding places. For example,
$c_{13}=a_{13}+b_{13}$. If two matrices have different sizes, their sum is not defined.
It is easy to prove that $A+B=B+A$, in other words the addition of matrices is commutative operation.

A $3\times 3$ matrix has $3$ columns and $3$ rows. For example, the sum of two $3\times 3$ matrices $A$ and $B$ is a matrix $C$ such that
$$\begin{align} C=&\left(
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right)+
\left(
\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} &b_{32} & b_{33} \\
\end{array}
\right)= \left(\begin{array}{ccc}
a_{11}+b_{11}& a_{12}+b_{12}& a_{13}+b_{13} \\
a_{21}+b_{21} &a_{22}+b_{22}& a_{23}+b_{23} \\
a_{31}+b_{31} &a_{32}+b_{32} & a_{33}+b_{33} \\
\end{array}\right)\end{align}$$
For example, let us find the sum for
$$A=\left(
\begin{array}{ccc}
10 & 20 & 10 \\
4 & 5 & 6 \\
2 & 3 & 5 \\
\end{array}
\right)\quad\mbox{and}\quad B=\left(
\begin{array}{ccc}
3 & 2 & 4 \\
3 & 3 & 9 \\
4 & 4 & 2 \\
\end{array}
\right)$$
Using the matrix addition formula, the sum of the matrices $A$ and $B$ is the matrix
$$A+B=\left(
\begin{array}{ccc}
10+3 & 20+2 & 10+4 \\
4+3 & 5+3 & 6+9 \\
2+4 & 3+4 & 5+2 \\
\end{array}
\right)=\left(
\begin{array}{ccc}
13 & 22 & 14 \\
7 & 8 & 15 \\
6 & 7 & 7 \\
\end{array}
\right)$$
The $3\times 3$ matrix addition work with steps shows the complete step-by-step calculation for finding the sum of two $3\times3$ matrices $A$ and $B$ using the matrix addition formula. For any other matrices, just supply elements of $2$ matrices in terms of a real numbers and click on the GENERATE WORK button. The grade school students may use this $3\times 3$ matrix addition to generate the work, verify the results of adding matrices derived by hand, or do their homework problems efficiently.

To represent data using a matrix, we will choose which category is represented by the columns and which category is represented by the rows. For example, it performed a study to find what types of sports are most popular and how they could attract more people. The matrix below represent their findings (percent of people)

**Practice Problem 1 : **

Find the sum of matrices $$X=\left(
\begin{array}{ccc}
-4 & 2 & 10 \\
-7 & 15 & -6 \\
12 & 0 & 5 \\
\end{array}
\right)\quad\mbox{and}\quad Y=\left(
\begin{array}{ccc}
-13 & 0 & -4 \
13 & 0 & 0 \\
45 & 8 & 2 \\
\end{array}
\right)$$
**Practice Problem 2 : **

Given triangle $\Delta ABC$ in three dimensional coordinate plane with $A(0,0,1),$ $B(3,6,2)$ and $C(-4,6,7)$. Translate this triangle for the vector $\vec a=(1,2,4)$.

The 3x3 matrix addition 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 addition of two or more matrices. Using this concept they can be able
to look at real life situations and transform them into mathematical models.