Determinant Calculator

Calculate the determinant of a matrix with detailed step-by-step explanations.

About Determinant Calculator

Welcome to our Determinant Calculator, a comprehensive tool designed to help you calculate the determinant of a matrix with detailed step-by-step explanations. This calculator is perfect for students, educators, and professionals dealing with linear algebra and matrix computations.

Features of the Determinant Calculator

Step-by-Step Solutions: Understand each step involved in calculating the determinant.
User-Friendly Interface: Easily input your matrix and get instant results.
Handles Various Matrix Sizes: Compute determinants of 1x1, 2x2, 3x3, and larger square matrices.

Understanding the Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It has important properties and applications in linear algebra, including solving systems of linear equations, finding the inverse of a matrix, and determining if a matrix is invertible.

Determinant of a 2x2 Matrix

For a 2x2 matrix:

\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]

The determinant is calculated as:

\[ \text{det}(A) = ad - bc \]

Determinant of a 3x3 Matrix

For a 3x3 matrix:

\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]

The determinant is calculated using the rule of Sarrus or the expansion by minors:

\[ \begin{align*} \text{det}(A) = & a_{11}(a_{22}a_{33} - a_{23}a_{32}) \\ & - a_{12}(a_{21}a_{33} - a_{23}a_{31}) \\ & + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \end{align*} \]

How to Use the Determinant Calculator

Enter your square matrix in the input field. Use new lines to separate rows and spaces or commas to separate elements.
Click on "Compute" to process your input.
View the determinant along with step-by-step solutions.

Applications of the Determinant

Solving Linear Systems: Determinants are used in Cramer's Rule to solve systems of linear equations.
Eigenvalues and Eigenvectors: Determinants are involved in finding eigenvalues of a matrix.
Area and Volume: Determinants can represent the scaling factor of linear transformations, affecting area and volume.
Invertibility: A matrix is invertible if and only if its determinant is non-zero.