Introduction to Matrix Arithmetic Tool
This is an online tool that helps students for performing the arithmetic operations (addition, subtraction, and multiplication) online. It is simple and easy to use. You just need to select the order of matrices. In Matrix Addition, Subtraction, and Multiplication Tool, you will fill the entries of those matrices and finally you can perform the operation of your choice.
If the order of matrices does not match, the operation can not be performed. For addition and subtraction, we know that two or more matrices should have the same order (equal number of rows and columns). Similarly, for multiplication we need to satisfy that the number of columns of first matrix should be equal to number of rows in the second matrix.
Matrix Arithmetic Tool
Example of Addition of mxn matrix
Here is an example of two matrices that are of order of mxn. When you added these matrices, the resultant matrix has also the same order.
\[
\begin{bmatrix}
a_{11} & a_{12} & \dots & a_{1n} \\
a_{21} & a_{22} & \dots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \dots & a_{mn}
\end{bmatrix}
+
\begin{bmatrix}
b_{11} & b_{12} & \dots & b_{1n} \\
b_{21} & b_{22} & \dots & b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \dots & b_{mn}
\end{bmatrix}
=
\begin{bmatrix}
a_{11}+b_{11} & a_{12}+b_{12} & \dots & a_{1n}+b_{1n} \\
a_{21}+b_{21} & a_{22}+b_{22} & \dots & a_{2n}+b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1}+b_{m1} & a_{m2}+b_{m2} & \dots & a_{mn}+b_{mn}
\end{bmatrix}
\]
\[
\mathbf{A} = \begin{bmatrix}
2 & 3 \\
4 & 5
\end{bmatrix}
,\quad
\mathbf{B} = \begin{bmatrix}
6 & 7 \\
8 & 9
\end{bmatrix}
\]
\[
\mathbf{A} + \mathbf{B} =
\begin{bmatrix}
2+6 & 3+7 \\
4+8 & 5+9
\end{bmatrix}
=
\begin{bmatrix}
8 & 10 \\
12 & 14
\end{bmatrix}
\]
Subtraction of Two matrices
We can perform the subtraction by subtracting the corresponding entries. In this case, the resultant still has the same order.
\[
\mathbf{A} = \begin{bmatrix}
5 & 8 & 3 & 6 \\
7 & 2 & 9 & 4 \\
1 & 0 & 5 & 3
\end{bmatrix}
,\quad
\mathbf{B} = \begin{bmatrix}
2 & 4 & 1 & 5 \\
3 & 1 & 7 & 2 \\
0 & 2 & 4 & 1
\end{bmatrix}
\]
\[
\mathbf{A} – \mathbf{B} =
\begin{bmatrix}
5-2 & 8-4 & 3-1 & 6-5 \\
7-3 & 2-1 & 9-7 & 4-2 \\
1-0 & 0-2 & 5-4 & 3-1
\end{bmatrix}
=
\begin{bmatrix}
3 & 4 & 2 & 1 \\
4 & 1 & 2 & 2 \\
1 & -2 & 1 & 2
\end{bmatrix}
\]
Multiplication of Two Matrices
\[
\mathbf{A} =
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix},
\quad
\mathbf{B} =
\begin{bmatrix}
7 & 8 \\
9 & 10 \\
11 & 12
\end{bmatrix}
\]
\[
\text{To compute } \mathbf{A} \times \mathbf{B}:
\]
\[
(1 \times 7) + (2 \times 9) + (3 \times 11) = 7 + 18 + 33 = 58
\]
\[
(1 \times 8) + (2 \times 10) + (3 \times 12) = 8 + 20 + 36 = 64
\]
\[
(4 \times 7) + (5 \times 9) + (6 \times 11) = 28 + 45 + 66 = 139
\]
\[
(4 \times 8) + (5 \times 10) + (6 \times 12) = 32 + 50 + 72 = 154
\]
\[
\therefore\quad \mathbf{A} \times \mathbf{B} =
\begin{bmatrix}
58 & 64 \\
139 & 154
\end{bmatrix}
\]
What is the Vector Algebra? addition, subtraction and multiplication of vectors