Matrix Determinant Calculator

The size of the matrix A:

Method:

Decomposition by:

Row number:

matrixA,0,0,matrixmatrixA,1,0,matrixmatrixA,2,0,matrixmatrixA,3,0,matrix

matrixA,0,1,matrixmatrixA,1,1,matrixmatrixA,2,1,matrixmatrixA,3,1,matrix

matrixA,0,2,matrixmatrixA,1,2,matrixmatrixA,2,2,matrixmatrixA,3,2,matrix

matrixA,0,3,matrixmatrixA,1,3,matrixmatrixA,2,3,matrixmatrixA,3,3,matrix

=Solve

About matrix determinant calculator

This is a free online matrix determinant calculator using Decomposition by row/column, Sarrus, Triangular form(Gaussian elimination), Montante (Bareiss algorithm) with complete, detailed, step-by-step description of solutions, that performs operations with matrices up to 99x99 in size with matrix elements of this type: decimal numbers, fractions, complex numbers, variables.

To start the calculation, you need to first enter the size of the matrix in the input field that you can find from the very top of the screen, also there you can choose the desired method of calculation.

A little below you will find a matrix window in which you need to enter matrix elements using the keyboard. The matrix control panel is also located here, which simplifies work with matrices and contains the following control elements:

The first element allows you to expand the matrix window. This can be especially useful in cases where you need to perform calculations with very large matrices that do not fit completely. If the matrix is still not visible after expanding the window, you can change the scale of the matrix using the + / - buttons;
The second element performs the function of copying the matrix input to the memory buffer. This can be useful in cases where you often use the same matrix for calculations, or if you need to move matrices between operations;
And the last element inserts the previously copied matrix, which allows you to speed up the process of entering the matrix to just a few clicks, instead of doing it manually;

And further down you will find a toolbar that allows you to customize the calculator and make it easier to work with it. It is visually divided into three parts, each of which is responsible for the following functionality:

The first allows you to select the number format when the solution result is displayed. Also, here you can turn off comments to the solution of the problem if you have already understood how to solve this problem, and you use the calculator to speed up or check your own calculations. Or you can turn off the step-by-step solution entirely if you only need the result of the solution;
The second contains buttons that allow you to change the type of the matrix input field, erase its elements or the entire matrix, and the largest button with an equal sign, which will take you to the screen with the solution of the problem. All these buttons are duplicated by keys on the keyboard. To find out which key on the keyboard to press, simply hover over one of the buttons and a tooltip will appear with the name of the key. You can also use the arrow keys on your keyboard to move the cursor between matrix input fields;
And the last one allows you to choose the number of digits after the decimal point for rounding non-integer numbers. Also, here you can immediately see an example of how rounded fractions will look;

What is matrix determinant?

The determinant of a matrix is a single scalar value that is a function of the elements of a square matrix and characterizes some properties of the matrix. So, the determinant of a matrix can be found only for square matrices, that is, those in which the number of columns and rows is the same. If the determinant of a matrix is zero, it means that the matrix is singular, also called degenerate or not invertible, and its inverse cannot be found.

How to find matrix determinant using Laplace expansion(Decomposition by certain row/column)?

Using the Laplace expansion, you can find the determinant of a square matrix of any size. To find the determinant of a matrix using Laplace expansion, also called cofactor expansion, first need to select any row or column of the matrix, usually this is the first row and further we will apply the explanation as if we had chosen the first row. Then you need to find the minor for each element in that row. To find the minor of some element, you need to remove a row and a column from the matrix that the element is in, this will give you a new submatrix for which you need to find the determinant, and this will give you the minor of that element. Then you need to find the cofactor for each element in a row by multiplying the minor of a certain element by 1 if the sum of the element's row index and column index is even, or -1 otherwise. Then you need to multiply each element in the row by its cofactor and sum all the resulting products, and the result will give you the determinant of the matrix.

Determinant calculation example

Write the initial matrix 

:

=

2

1

0

-1

4

2

3

0

5

To find the determinant of matrix 

 need to do the following:

Find the minors for each element of 1th row of the matrix A;

Multiply each element of 1th row of the matrix A by its corresponding minor;

The product of an element in its minor must be taken with a minus or plus sign, it depends on the indices of the element:

if i + j is equal to an even number, then take the plus sign;

if i + j is not an even number, then take the minus sign;

Add all the products of the element on the corresponding minor, taking into account the correct taken sign;

det(

) =

= 1

(-1)

i+j

·

i,j

·

i,j

// where

is the row number

is the column number

det(

) =

2

1

0

-1

4

2

3

0

5

=

((

-1

(1+1)

*

1,1

)

*

1,1

)

  +  ((

-1

(1+2)

*

1,2

)

*

1,2

)

  +  ((

-1

(1+3)

*

1,3

)

*

1,3

)

M1,1

1,1

=

2

1

0

-1

4

2

3

0

5

=

4

2

0

5

=

20

;

M1,2

1,2

=

2

1

0

-1

4

2

3

0

5

=

1

0

0

5

=

5

;

M1,3

1,3

=

2

1

0

-1

4

2

3

0

5

=

1

0

4

2

=

2

;

Matrix determinant

det(

) =

(

1

*

2

*

20

)

+

(

-1

*

-1

*

5

)

+

(

1

*

3

*

2

)

=

51

;

Answer

det(A)

det(

) =

51

;

SIZE3×3METHODCofactor expansion (Laplace)

Frequently asked questions

How do you find the determinant of a 3×3 matrix?

Expand along any row or column using cofactor expansion: multiply each entry by its signed minor and add the results. For a 3×3 matrix you can also use the Sarrus rule, which sums the products of the three forward diagonals and subtracts the products of the three backward diagonals.

What does a determinant of 0 mean?

A determinant of 0 means the matrix is singular: its rows (and columns) are linearly dependent, it has no inverse, and the linear system it represents has either no solution or infinitely many solutions.

Can a non-square matrix have a determinant?

No. The determinant is defined only for square matrices, where the number of rows equals the number of columns. For non-square matrices, related quantities such as the rank or singular values are used instead.

What is the determinant used for?

The determinant shows whether a matrix is invertible, measures how the matrix scales area or volume, appears in Cramer's rule for solving linear systems, and is used to find eigenvalues through the characteristic polynomial.

Calculation methods

Sources

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