# Online Calculator for The Determinant of a Matrix of Any Size

An online calculator to calculate the determinant of a square matrix is presented. An interactive tutorial related to linerarly independent columns and rows of a matrix is suggested.

## How to Use the Calculator?

Enter the number of columns (and rows) \( n \) below, click on "Generate Matrix" to generate a matrix with random values of its elements. You may change the values of the elements by entring new values and click on "Update Matrix". You may enter the values of the elements of the matrix as integers, decimal numbers such as 1.2 or fractions such as -4/5.The steps per column are shown: In blue the row echelon form and in red the row reduced form.

Enter Number of columns (and rows) \( n = \)

Click here to enter \( n \) and generate a matrix whose elemenst have random values

Change values of elements in the above matrix (if needed) and click

## Interactive Tutorial

- The determinant of a matrix with linearly dependent rows is equal to zero.

Enter matrices with linearly dependent rows, in the calculator, and check that the above is true. - The determinant of a matrix with linearly dependent columns is equal to zero.

Enter matrices with linearly dependent columns , in the calculator, and check that the above is true.

- A matrix is invertible if and only if its determinant is NOT equal to zero.

Use the calculator to determine which of the following matrices are invertible.

a) \( \begin{bmatrix} 2 & 1 & -1 \\ 3 & 1 & 0 \\ -1 & 0 & 2 \end{bmatrix} \) b) \( \begin{bmatrix} -5 & 0 & -1 & 7\\ 2 & 2 & 0 & 9 \\ -6 & 4 & -2 & 32 \\ -8 & 0 & 2 & 1 \end{bmatrix} \) c) \( \begin{bmatrix} -5 & 1 & -1 & 7 & 5\\ -4 & 1 & 0 & 9 &-2 \\ -1 & 2 & -2 & 7 & 0\\ -8 & 0 & 2 & 1 & -1\\ -9 & 2 & -1 & 16 & 3 \end{bmatrix} \) d) \( \begin{bmatrix} -15 & 12 & -1 & 7 & 5 & 8\\ -4 & 11 & 0 & 9 &-2 & 5\\ 2 & 2 & -2 & 7 & 0 & 2\\ -8 & -2 & 12 & 1 & -1 & -3\\ -7 & 2 & -1 & 0 & 3 & 2 \\ 0 & 4 & -4 & -1 & 3 & -3 \end{bmatrix} \)