# Matrix Calculator

Matrix calculator allows you to find the matrix addition, subtraction, multiplication, transpose, inverse, and determinant online.

Matrix A Input
 row column ×
Matrix B Input
 row column ×

## About Matrix Calculator

Using this online matrix calculator, you can easily find the solution for your matrix problems. It supports almost all the operations. You can add, subtract, or multiply matrices, find their inverse, calculate determinants, and so on. In short, you can say it is a one-stop destination for all the operations. Also, it combines all the different calculators like matrix multiplication calculator, inverse matrix calculator, determinant calculator, and much more.

### What is Matrix?

A matrix can be simply defined as a set of numbers that are arranged in rows and columns to form a rectangular array. To form a matrix, there should be at least one column or row. Also, there are different kinds of matrix and they are widely used in mathematical calculations, statistical implementations, physics, economics, computer science, and economics.

#### Dimensions

The dimensions of a matrix tell about the number of rows and columns of the matrix. It is denoted as m x n, where m is the number of rows and n is the number of columns.

#### Representation

Let's consider a 2 × 3 matrix A. So, it can be represented as A2 x 3. It includes 2 rows and 3 columns.

The elements of the matrix can be represented in an array form as follows:

A=
 4 2 3 5 6 1

Each element in the above matrices represented as:

A1 , 1 = 4
A1 , 2 = 2
A1 , 3 = 3
A2 , 1 = 5
A2 , 2 = 6
A2 , 3 = 1

This is the way to represent a matrix as individual entities. The first subscript denotes row and the second one denotes column. Using this value, you can determine the position of an element easily.

#### Features of Matrix Calculator

The matrix calculator gives so many features in it using which you can easily find solutions to your problems.

• You can set the number of rows and columns to operate on the different matrices.
• All the operations can be performed using the buttons.
• Operations like transpose, determinant, power, and inverse which are related to an individual matrix are present with the dedicated matrix.
• Whereas the operations like addition, subtraction, multiplication, and reverse are present out of the box.
• Also, there are some extra buttons to make your work easy. You can use these extra utility buttons like Clear, All 0, All 1, Random, and Power of to ease your matrix calculations.
• Simple and clear notations makes it easy to understand the tool.
• The tool works on pre-defined algorithms. So, it gives you 100% accurate results quickly.
• Simple and user-friendy interface. Therefore, it's super easy to operate.
• Our tool works on almost all the browsers and all small and large deivces.

1. Saves your time and effort.
2. Free to use.
3. No usage limit. Use unlimited times.
4. 100% accurate. No chances of error.
5. You can use it anytime, anywhere.
6. All major matrix operations supported.
7. Easy to understand and very lightweight.

### How to use Matrix Calculator?

You can easily use our matrix calculator to perform all the matrix-related operations and the basic steps for all of them remain the same. Just follow the instructions below.

1. Firstly, Start by setting the number of rows and columns of both the matrices. Also, you can simply press the up-down and right-left arrows to increase or decrease the rows and columns.
2. Then fill the matrix with the correct value at the correct position.
3. Now, just click on the operation to perform. To add both the matrices click on the "A + B" button. Similarly, you can press the "A – B" or "AB" button to subtract or multiply both matrices. "A ↔ B" button will swap two matrices.
4. Also, there are some more buttons that are used to find the transpose, determinant, inverse, and power of the matrix. You just need to enter the values and click the button which you want to find. That's it.
5. The resultant values will be shown at the bottom of the calculator.
6. "All 0" button will add all the values as 0 in all the fields. Similarly, "All 1" is used to add all values as 1.
7. When you click on the "Random" button, it will take random values from 0 to 9 and fill the full matrix with random values.
8. Lastly, you can make empty the full matrix using the "Clear" button.

## Matrix Operations

This is the main context that we are going to talk about in this article. Matrix operations are the set of operations that we can apply to find some results. The matrix calculator makes your task easy and fast. Also, you can perform these operations with just a few keystrokes. The most common matrix operations are addition, subtraction, multiplication, power, transpose, inverse, and calculating determinant. Let's learn all of them one by one.

Matrix addition is only performed if both matrices are the same size. In other words, the matrices to be added should have the exact same number of rows and columns. This means they should have the same dimensions.

Addition Process: When the size/dimensions of the matrices are the same then to find the sum, you need to add the corresponding elements of the matrices.

Formula: Ci , j = Ai , j + Bi , j

Here, A and B are the elements of the matrices to be added and C is the resultant matrix.

##### Example:
A =
 4 3 6 5
B =
 5 3 2 7
C =
 4 + 5 3 + 3 6 + 2 5 + 7
C =
 9 6 8 12

### Matrix Subtraction

Matrix subtraction is very much similar to addition. It is also performed if both matrices are the same size. In other words, both matrices should have exact same number of rows and columns.

Subtraction Process: You need to subtract the corresponding elements of the matrices.

Formula: Ci , j = Ai , j - Bi , j

Let's understand it from a simple example.

##### Example:
A =
 24 17 9 12
B =
 12 8 2 18
C =
 24 - 12 17 - 8 9 - 2 12 - 18
C =
 12 9 7 -6

### Matrix Multiplication

There are two types of matrix multiplication we can perform.

1. Scalar multiplication
2. Matrix multiplication with other matrix

#### 1. Scalar multiplication

In scalar multiplication, each element of the single matrix is multiplied by a scalar value. Let's take an example to understand it.

##### Example:
A =
 2 4 5 6
B = 3
3 ×
 2 4 5 6
=
 6 12 15 18

#### 2. Matrix multiplication with other matrix

To multiply two matrices, the number of columns of the first matrix should be equal to the number of rows of the second matrix.

For example, you can multiply 3 × 4 with 4 × 2. But you can not multiply 3 × 4 with 2 × 4.

Note: Matrix multiplication is not a commutative property. It means, it is not necessary that A × B will always be equal to B × A.

Multiplication Process: Two matrices are multiplied by finding the dot product between the corresponding elements of the row of the first matrix and the column of the second matrix.

Formula:
C1 , 1 = (A1 , 1 , A1 , 2) × (B1 , 1 , B2 , 2)
Or
C1 , 1 = A1 , 1 × B1 , 1 + A1 , 2 × B2 , 2

Here, A and B are the elements of the matrices to be multiplied and c is the resultant matrix.

##### Example:
A =
 4 2 3 5
B =
 1 6 0 5
C =
 (4 × 1)+(2 × 0) (3 × 1)+(5 × 0) (4 × 6)+(2 × 5) (3 × 6)+(5 × 5)
C =
 4 34 3 43

### Power of Matrix

Power of matrix meaning is to raise a matrix to a given power.

For example, the given matrix is A. Now calculate matrix A with the power of 2 means: A2 = A × A.

Most importantly, the power can be raised to only square matrices. Because a non-square matrix can not be multiplied by itself.

##### Example:
A2 =
 4 8 3 2
2
=
 4 8 3 2
×
 4 8 3 2
=
 40 48 18 28

### Matrix Transpose

The transpose of a matrix flips its elements over its diagonal. The row elements become column elements whereas the column elements become row elements. Most importantly, the matrix should not be empty.

Transpose Process: To transpose a matrix just switch the rows and column elements. If the matrix contains 2 rows and 3 columns the matrix will now consist of 3 rows and 2 columns.

Formula: Tm , n = Tn , m = T'

Here, T is a matrix containing m rows and n columns that will become n rows and m columns after transpose.

##### Example:
A =
 6 3 9 7
AT =
 6 9 3 7
B =
 10 12 15 20 9 18 22 41
BT =
 10 9 12 18 15 22 20 41

### Matrix Determinant

The Laplace formula is widely used to calculate the determinant of the matrix of any size. Let's understand the process through this formula and example:

##### Formula:
|A|=
 a b c d e f g h i
=
 e f h i
-   b
 d f g i
+   c
 d e g h

= a(ei-fh) - b(di-fg) + c(dh-eg)

##### Example:
|A|=
 3 7 3 5 1 3 2 5 4
=
 1 3 5 4
-   5
 5 3 2 4
+   2
 5 1 2 5

= 3(1×4 - 5×3) – 5(7×4 – 5×3) + 2(7×3 – 3×1)

= 3(-11) – 5(13) + 2(18)

= -62

### Matrix Inverse

Pre-requisites: The matrix should not be empty and you should know the determinant of that matrix. Also, the determinant should not be equal to zero.

Process: To find the inverse of the matrix we use a simple formula where the inverse of the determinant is multiplied with the adjoint of the matrix.

Formula: A-1 = ( 1 / |A| ) × adj(A)

Where, the adjoint of a matrix is the collection of its cofactors which are the determinants of the minor matrices.