Matrix

Do you need a matrix calculator? We have a wide selection of them so that you can solve different operations such as calculating the matrix inverse, the inverse or the multiplication of matrices.

Matrix operations

Choose the matrix calculator you need below:

• Transpose Matrix Calculator
• Calculate 2×2 inverse matrix
• Calculate 3×3 inverse matrix
• 4×4 Inverse Matrix Calculator
• Multiplication of 2×2 matrices
• Multiplying 3×3 matrices

Do you need any other matrix calculator? Ask us and we will do it. We also remind you that if you want to calculate a determinant, in the link we just left you can solve them online.

In addition, in each of the matrix calculators that we have left you to solve operations, you will find how to do it with the formula and solved examples so that you understand how it is done in each case. Of course, if you have any questions you can ask us and we will be happy to help you.

Types of matrices

There are many types of matricesTherefore, we will summarize below which are the most important ones.

Row matrix

The row matrix is the one that consists of a single row as its name indicates. An example of a row matrix can be the following:

Column matrix

In the case of the column matrix, as its name indicates, we have a single column. An example of this type of matrix is:

Rectangular matrix

We say that a matrix is rectangular when the number of rows and columns is differentthat is, its dimension is mxn. For example, below you can see a 3×2 rectangular matrix:

Square matrix

Yes the number of rows and columns is the samethen it is a square matrix. That is, its dimension is nxn as in the following example:

Within the square matrices there are several types such as upper triangular, lower triangular, diagonal, scalar, identity, orthogonal and more. Here you can see all the types of square matrices there are.

Transposed matrix

The transpose matrix (A_t) of a matrix A is the one in which we have exchanged the rows for columns. For a better understanding, below you can see the formula of the transpose matrix.

If you want to know more about how it is calculated, its properties or see practical examples, do not hesitate to visit our matrix transpose calculator.

Null matrix

The null matrix is characterized by all its elements are 0.

By its characteristic, the null matrix has the following properties:

• It is symmetrical and antisymmetrical
• It is unique
• It is nilpotent

Types of square matrices

As there is a great variety of square matrices, we wanted to dedicate a special section to talk a little bit about each one of them so that you can learn to identify them well.

As we have said before, a square matrix is one that has the same number of rows and columns. Now we will see what types there are:

Upper triangular matrix

The upper triangular matrix is the one whose elements below the main diagonal are zero.

Lower triangular matrix

In the lower triangular matrix, just the opposite of the previous type of matrix occurs, i.e., in this case all the elements above the main diagonal are zero.

Diagonal matrix

The diagonal matrix is the one that only has elements on its main diagonal. The rest will be zero.

Among its properties, it is worth noting that the diagonal matrix is square and symmetricala.

Scalar matrix

The scalar matrix is like the diagonal matrix but differs from the diagonal matrix in that all the elements of the main diagonal are equal.

Matrix identity or unit

Yes the elements of the main diagonal are all equal to 1we are in front of the identity matrix.

The identity matrix is very important because acts as a neutral element in the multiplication of matricesthat is, if we multiply any matrix by the unit matrix, the original matrix does not change.

Regular matrix

We say that it is a regular matrix when we can calculate its inverse. If you don't know how to do it, here are a couple of links to show you how to calculate the inverse matrix of rank 2×2 or 3×3:

• Calculate 2×2 inverse matrix
• Calculate 3×3 inverse matrix

Singular matrix

If, on the other hand, the matrix has no inverseis a singular matrix. For this to be true, its determinant must be zero.

Idempotent matrix

A matrix is idempotent when when squared, we are left with the original matrix. This mathematically expressed is represented as follows:

A² = A

Here are some of them examples of idempotent matrices in which the previous property can be demonstrated:

Involute matrix

We will say that a matrix is involutive if by multiplying it by itself we obtain the identity matrix. That is to say:

A² = I

Below are a couple of solved exercises of involutive matrices in which you can see how the above expression is fulfilled:

Among the properties of the involute matrix we have to:

• They are square matrices
• They are inverses of themselves

Symmetric matrix

A symmetric matrix is a matrix that is equal to its transpose matrixi.e:

A = A^t

Antisymmetric matrix

We also have a type of matrix that is called antisymmetrical and that is equivalent to its transpose matrix but with a minus sign in front of:

A = -A^t

Among the properties of the antisymmetric matrix we have:

• It is a square matrix
• The elements of the main diagonal are equal to zero.

Orthogonal matrix

To finish with the types of matrices we are going to see the orthogonal matrix. It is so called because if we multiply it by its transpose, we obtain the identity matrix:

A - A^t = I

For example:

Our matrix calculators

As you may have seen, we have several matrix calculators whose operation is very simple.

You only have to select the matrix calculator for the operation you want to perform and you will see that there is an nxm range grid.

In each of the cells you will have to write the elements of the matrix on which you want to operate, so that when you have done so you only have to press the calculate button to obtain the result.

Do you want to calculate determinants online or know what are its properties? Below you have all the information you are looking for to solve determinants of a matrix.

Calculate determinants online

Do you have to calculate the determinant of a matrix? Here you will find a collection of online calculators to solve different types of determinants depending on their size:

• 2×2 determinant
• 3×3 determinant
• Determinant 4×4

In all cases we will show you how to solve the determinanteither through its mathematical method or through Excel.

Properties of determinants

It is convenient to to know the properties of determinants to solve them correctly and not to make mistakes when solving exercises in which they are involved.

1. The determinant of A is equal to the determinant of its transpose

If you calculate the determinant of the matrix A and then do the same for the determinant of its transposed matrixyou will see that they are worth exactly the same. Therefore, you have this equality:

|A^t|= |A|

2. The determinant of a matrix is 0 if:

|A| can be equal to zero if some of these conditions are met:

• Has two rows or columns that are exactly the same
• All the elements of a row or of one of its columns are null, that is, equal to zero.
• The elements of a row or column are a linear combination of the others. For example, it can be the case that adding the elements of row 1 and row 2 results in the numbers of row 3, so in that case the determinant will be zero. A clear example of this criterion is the one that can be seen in the following example:

If you look at the determinant above, the sum of row 1 and row 2 results in row 3, so they are a linear combination and their result is zero.

3. A triangular determinant is equal to the product of the elements of its main diagonal.

A triangular determinant is the one that is full of zeros above or below the main diagonal. This allows us to calculate how much it is worth by simply multiplying the elements placed on the main diagonal.

4. When two rows or two columns are switched between them, the value of the determinant changes sign.

As can be seen in the following example, if we move two rows or two columns of the determinant with respect to each other, its final value is the same but we have to change the sign.

5. If you convert a row or column into a linear combination of the others, the value of the determinant does not change.

The only condition for this property to be fulfilled is that the following must be present multiplying the elements of one of the row or column of the determinant by a number real.

In the following example you can see how if we convert Column 3 into a linear combination of column 1 and 2, we have as a result the same value. However, column one has been multiplied by two previously.

1. When multiplying a determinant by a real number, we have to multiply only one of its rows or columns.

That is, if we multiply a determinant by 2, we will only multiply by that number one of its rows or one of its columns.

7. If the elements of a row or column are formed by two summands, that determinant can be decomposed in two.

In this case it is important to note that the remaining rows or columns of the determinant that have only a single summand will remain unchanged in each of the resulting determinants.

In this example, the theory explained in this point can be seen much more clearly:

8. |A - B| = |A| - |B|

To finish the properties of determinants, if we have the determinant of a product can be equated to the product of determinants.