With this calculator, you can perform matrix addition, subtraction, or multiplication, as well as calculate the determinant of a matrix. To perform the necessary calculations, you need to sequentially click on the buttons indicated under the example in the corresponding section.
Transposing a matrix can be easily done even without a calculator, simply follow the detailed instructions.
What is a Matrix in Mathematics
A matrix is a rectangular array of elements (numbers, symbols, or expressions), consisting of $m$ rows and $n$ columns. Each element of the matrix is located at the intersection of a specific row and column.
A matrix is usually denoted by a capital letter, for example, $A$. Individual elements of the matrix are denoted using indices, for example, $a_{12}$ is the element located in the first row and second column.
$$A = \begin{bmatrix}a_{11} & a_{12} & \dots & a_{1n} \\a_{21} & a_{22} & \dots & a_{2n} \\\vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & \dots & a_{mn}\end{bmatrix}$$
The size of a matrix is denoted as $m \times n$. For example, a $3 \times 4$ matrix will have 3 rows and 4 columns. The number of elements in a matrix can be found by multiplying $m$ by $n$ on a regular calculator ($3 \cdot 4 = 12$).
Matrix Addition and Subtraction
Matrix addition and subtraction are operations in which corresponding elements of the matrices are added or subtracted. It is necessary that the matrices have the same size, that is, they have the same number of rows and columns.
Example of matrix addition:
$$\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6\end{bmatrix}\ + \\begin{bmatrix}1 & 2 & 3 \\3 & 2 & 1\end{bmatrix} \ = \$$
$$\ = \ \begin{bmatrix}1+1 & 2+2 & 3+3 \\4+3 & 5+2 & 6+1\end{bmatrix} \ = \$$
$$\ = \ \begin{bmatrix}2 & 4 & 6 \\7 & 7 & 7\end{bmatrix}$$
2nd [ [ 1 , 2 , 3 ]
[ 4 , 5 , 6 ] ]
+ [ [ 1 , 2 , 3 ]
[ 3 , 2 , 1 ] ] =
Example of matrix subtraction:
$$\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6\end{bmatrix}\ - \\begin{bmatrix}3 & 2 & 1 \\1 & 2 & 3\end{bmatrix} \ = \$$
$$\ = \ \begin{bmatrix}1-3 & 2-2 & 3-1 \\4-1 & 5-2 & 6-3\end{bmatrix} \ = \$$
$$\ = \ \begin{bmatrix}-2 & 0 & 2 \\3 & 3 & 3\end{bmatrix}$$
2nd [ [ 1 , 2 , 3 ]
[ 4 , 5 , 6 ] ]
- [ [ 3 , 2 , 1 ]
[ 1 , 2 , 3 ] ] =
Matrix Multiplication
Multiplication of two matrices is an operation of calculating a new matrix, which is called the product of matrices. Each element of this matrix is equal to the sum of the products of the elements in the corresponding row of the first matrix and the column of the second matrix. For matrix multiplication, it is necessary that the number of columns in the first matrix is equal to the number of rows in the second matrix.
Example of matrix multiplication:
$$\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6\end{bmatrix}\ \cdot \\begin{bmatrix}7 & 8 \\9 & 10 \\11 & 12\end{bmatrix} \ = \$$
$$\ = \ \begin{bmatrix}1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \\4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12\end{bmatrix} \ = \$$
$$\ = \ \begin{bmatrix}58 & 64 \\139 & 154\end{bmatrix}$$
2nd [ [ 1 , 2 , 3 ]
[ 4 , 5 , 6 ] ]
2nd × 2nd [ [ 7 , 8 ]
[ 9 , 1 0 ]
[ 1 1 , 1 2 ] ] =
Matrix Determinant
The determinant of a matrix ($det(A)$ or $|A|$) is a value that characterizes the properties of a square matrix.
Example of calculating the determinant (det must be entered in the empty field under the calculator screen, using your computer’s keyboard):
$$|A| = \begin{vmatrix}1 & -2 & 3 \\4 & 0 & 6 \\-7 & 8 & 9\end{vmatrix} = 204$$
d e t ( 2nd [
[ 1 , - 2 , 3 ]
[ 4 , 0 , 6 ]
[ - 7 , 8 , 9 ]
] 2nd ) =
Matrix Transposition
Transposition is an operation in which the rows and columns of the original matrix are interchanged, that is, the rows become columns, and the columns become rows. If $A$ is the original matrix, then the transposed matrix is denoted as $A^T$. If the original matrix $A$ has a size of $m \times n$, then the transposed matrix $A^T$ will have a size of $n \times m$.
$$A = \begin{bmatrix}a_{11} & a_{12} & \dots & a_{1n} \\a_{21} & a_{22} & \dots & a_{2n} \\\vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & \dots & a_{mn}\end{bmatrix}$$
To obtain the transposed matrix $A^T$, you need to interchange the rows and columns of the original matrix $A$. To do this, you need to perform the following steps.
Take the elements of the first row from $a_{11}$ to $a_{1n}$ and write them as the first column of the transposed matrix $A^T$:
$$A^T = \begin{bmatrix}a_{11} \\a_{12} \\\vdots \\a_{1n}\end{bmatrix}$$
Take the elements of the second row from $a_{21}$ to $a_{2n}$ and write them as the second column of $A^T$:
$$A^T = \begin{bmatrix}a_{11} & a_{21} \\a_{12} & a_{22} \\\vdots & \vdots \\a_{1n} & a_{2n}\end{bmatrix}$$
Repeat this step for all rows of the matrix $A$ until they are written as columns of $A^T$:
$$A^T = \begin{bmatrix}a_{11} & a_{21} & \dots & a_{m1} \\a_{12} & a_{22} & \dots & a_{m2} \\\vdots & \vdots & \ddots & \vdots \\a_{1n} & a_{2n} & \dots & a_{mn}\end{bmatrix}$$
Thus, the elements $a^T_{ij}$ of the transposed matrix $A^T$ correspond to the elements $a_{ji}$ of the original matrix $A$.
Example of matrix transposition:
$$A = \begin{bmatrix}1 & 2 \\3 & 4 \\5 & 6\end{bmatrix}\ \ A^T = \begin{bmatrix}1 & 3 & 5 \\2 & 4 & 6\end{bmatrix}$$
