Matrix determinant calculator

  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.

  How to find matrix determinant using the Rule of Sarrus?

The Rule of Sarrus can be applied only to matrices of size 3 x 3. In order to find the determinant using the Rule of Sarrus, you first need to write out the first two columns of the matrix to the right of the third column, thus obtaining a matrix with five columns. Then you need to add the products of the diagonals going from top to bottom and subtract the products of the diagonals going from the bottom to the top and the result will be the determinant of the matrix.

  How to find matrix determinant using a Triangular form(Gaussian elimination)?

Using the Triangular form, you can find the determinant of a square matrix of any size. In order to find the determinant of a matrix, we can use the property of triangular matrices, which says that the determinant of a triangular matrix is the product of the elements of its main diagonal. So, first you need to use the Gaussian elimination to bring the matrix to a triangular form and then multiply all the elements on the main diagonal and the result will be the determinant of the matrix.

  How to find matrix determinant using Montante (Bareiss algorithm)?

Using the Montante (Bareiss algorithm), you can find the determinant of a square matrix of any size. To find the determinant of a matrix, you just need to apply the Bareiss algorithm to the matrix, which will bring it to echelon form, and then the last element on the main diagonal will be the determinant of the matrix.