Determinant Calculator
Calculate the determinant of any square matrix with detailed step-by-step cofactor expansion, matrix properties analysis, and visual solutions.
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About Determinant Calculator
Welcome to the Determinant Calculator, a professional-grade tool for computing matrix determinants with comprehensive step-by-step solutions. Whether you are studying linear algebra, solving systems of equations, or analyzing matrix properties, this calculator provides detailed cofactor expansion breakdowns and matrix insights.
What is a Determinant?
The determinant is a scalar value computed from the elements of a square matrix. It encodes fundamental information about the matrix and the linear transformation it represents. The determinant has deep geometric and algebraic significance in mathematics.
Key Properties of Determinants
A non-zero determinant indicates the matrix is invertible. The absolute value represents the scaling factor of volumes under the transformation. The sign indicates orientation preservation or reversal.
2x2 Determinant Formula
For a 2x2 matrix, the determinant is calculated directly:
3x3 Determinant Formula
For a 3x3 matrix, use cofactor expansion along any row or column:
Where each cofactor $C_{ij} = (-1)^{i+j} \cdot M_{ij}$ and $M_{ij}$ is the minor (determinant of the submatrix with row i and column j removed).
How to Use This Calculator
- Select matrix size: Choose from 2x2 to 6x6 using the size buttons, or enter any square matrix in the text area.
- Enter values: Fill in the interactive grid or type values directly. Use spaces or commas to separate elements, new lines for rows.
- Calculate: Click the Calculate button to compute the determinant.
- Review solution: Examine the step-by-step cofactor expansion showing all intermediate calculations.
- Check properties: Review the matrix properties panel to understand invertibility and other characteristics.
Applications of Determinants
Solving Linear Systems (Cramer's Rule)
Determinants enable direct solution of linear equation systems. For Ax = b, each variable can be expressed as a ratio of determinants.
Matrix Invertibility
A matrix is invertible if and only if its determinant is non-zero. This is fundamental in checking whether systems have unique solutions.
Eigenvalue Problems
Eigenvalues are found by solving det(A - λI) = 0, the characteristic polynomial. This is central to many applications in physics and engineering.
Geometric Transformations
The determinant gives the signed volume scaling factor. A determinant of 2 doubles areas/volumes; a negative determinant indicates reflection.
Understanding Matrix Properties
Singular vs Invertible Matrices
- Invertible (det ≠ 0): The matrix has a unique inverse, rows/columns are linearly independent, and systems Ax = b have unique solutions.
- Singular (det = 0): The matrix has no inverse, rows/columns are linearly dependent, and systems may have no solution or infinitely many.
Trace and Determinant Relationship
The trace (sum of diagonal elements) and determinant are related through eigenvalues. For a matrix with eigenvalues λ₁, λ₂, ..., λₙ:
- Trace = λ₁ + λ₂ + ... + λₙ
- Determinant = λ₁ × λ₂ × ... × λₙ
Frequently Asked Questions
What is the determinant of a matrix?
The determinant is a scalar value computed from a square matrix that encodes important properties. It indicates whether a matrix is invertible (non-zero determinant), represents the scaling factor of linear transformations, and equals the signed volume of the parallelepiped formed by row/column vectors.
How do you calculate a 2x2 determinant?
For a 2x2 matrix [[a,b],[c,d]], the determinant is calculated as det = ad - bc. Multiply the main diagonal elements (a×d), subtract the product of the anti-diagonal elements (b×c).
How do you calculate a 3x3 determinant?
For a 3x3 matrix, use cofactor expansion along any row or column. Expand along the first row: det(A) = a₁₁·C₁₁ + a₁₂·C₁₂ + a₁₃·C₁₃, where each cofactor Cᵢⱼ is (-1)^(i+j) times the determinant of the 2x2 minor matrix.
What does a zero determinant mean?
A zero determinant indicates the matrix is singular (non-invertible). This means the rows/columns are linearly dependent, the matrix maps some non-zero vector to zero, and the system of equations Ax=b either has no solution or infinitely many solutions.
Can you calculate the determinant of a non-square matrix?
No, determinants are only defined for square matrices (same number of rows and columns). For non-square matrices, related concepts like pseudo-determinants or singular values can be computed, but the classical determinant does not exist.
What is cofactor expansion?
Cofactor expansion (Laplace expansion) calculates a determinant by expanding along any row or column. For each element aᵢⱼ, multiply it by its cofactor Cᵢⱼ = (-1)^(i+j) × Mᵢⱼ, where Mᵢⱼ is the minor (determinant of the submatrix with row i and column j removed). Sum all products to get the determinant.
Additional Resources
Reference this content, page, or tool as:
"Determinant Calculator" at https://MiniWebtool.com/determinant-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: Jan 17, 2026
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