Determinant Calculator Online | Matrix Inverse Calculator
Free online determinant calculator and matrix inverse calculator with detailed step-by-step solutions. This essential linear algebra tool helps students, engineers, and mathematicians solve complex matrix operations, calculate determinants of square matrices, and find matrix inverses for solving systems of linear equations and understanding linear transformations.
Determinant & Inverse Calculator
Matrix Determinant
The determinant is a scalar value that can be computed from a square matrix and encodes certain properties of the linear transformation described by the matrix.
For a 2×2 matrix:
det(A) = ad - bc
where A = [a b; c d]
Key properties of determinants:
If det(A) ≠ 0, the matrix is invertible
If det(A) = 0, the matrix is singular
det(AB) = det(A) × det(B)
det(A⁻¹) = 1/det(A)
Matrix Inverse
The inverse of a square matrix A is another matrix A⁻¹ such that A × A⁻¹ = A⁻¹ × A = I, where I is the identity matrix.
For a 2×2 matrix:
A⁻¹ = (1/det(A)) × [d -b; -c a]
where A = [a b; c d]
Key facts about matrix inverses:
Only square matrices with non-zero determinants have inverses
If A is invertible, the solution to Ax = b is x = A⁻¹b
(AB)⁻¹ = B⁻¹A⁻¹
Typical Use Cases
Our determinant calculator online and matrix inverse calculator serve students, engineers, and researchers working with advanced mathematical computations. Mathematics students use this linear algebra tool for homework verification, learning matrix operations concepts, and solving complex determinant and inverse problems with step-by-step solutions.
Engineering professionals rely on this matrix operations calculator for structural analysis, control systems design, and signal processing applications. Computer science researchers use this matrix math tool for machine learning algorithms, computer graphics transformations, and cryptographic systems requiring precise matrix computations.
Practical Applications
Engineering
Structural analysis
Circuit analysis
Control systems
Signal processing
Computer Science
Computer graphics
Machine learning
Cryptography
Network analysis
Physics
Quantum mechanics
Electromagnetism
Mechanics
Relativity
Calculation Methods
Various algorithms can be used to calculate determinants and inverses, each with different computational efficiency and numerical stability characteristics:
Determinant Methods
Cofactor expansion
Recursive method based on minors and cofactors
Gaussian elimination
Transforms matrix to triangular form, product of diagonal
LU decomposition
Factorizes matrix as product of lower and upper triangular matrices
Inverse Calculation
Adjugate method
Uses adjugate matrix and determinant
Gauss-Jordan elimination
Row operations to transform matrix to identity
Numerical methods
Iterative approaches for large matrices
Example Code (Python with NumPy)
import numpy as np
# Create a matrix
A = np.array([[4, 7], [2, 6]])
# Calculate determinant
det_A = np.linalg.det(A)
print(f"Determinant: {det_A}") # Output: 24.0 - 14.0 = 10.0
# Calculate inverse if determinant is not zero
if det_A != 0:
A_inv = np.linalg.inv(A)
print(f"Inverse matrix:\n{A_inv}")
else:
print("Matrix is not invertible")