Matrix Calculator — 2×2 & 3×3 Matrix Solver, Determinant & MultiplicationAdd · Subtract · Multiply · Determinant · Inverse · Transpose · Linear Algebra
Use this free Matrix Calculator to instantly perform all standard matrix operations for 2×2 and 3×3 matrices — the most commonly used matrix dimensions in linear algebra, engineering mathematics, and physics. Supported operations include: Matrix Addition (A + B) · Matrix Subtraction (A − B) · Matrix Multiplication (A × B) · Determinant calculation det(A) · Matrix Inverse (A⁻¹) — where det(A) ≠ 0 · Matrix Transpose (Aᵀ) — with full step-by-step working shown for each operation, making it ideal for both quick matrix verification and learning matrix arithmetic.
This online matrix solver is trusted across every level of mathematics and engineering education: A-Level, GCSE, SAT, AP Mathematics, JEE, and NEET linear algebra problems, university linear algebra and matrix theory coursework, engineering mathematics — circuits, structural analysis, and control systems, computer graphics — transformation matrices and 3D rotation, machine learning — weight matrices and linear transformations, and physics — quantum mechanics and tensor analysis. Key matrix concepts covered include: singular vs non-singular matrices, identity matrix and zero matrix properties, cofactor expansion for determinant calculation, Cramer's Rule for solving linear equations, and matrix rank and linear independence — making this the most comprehensive free matrix calculator online for students, engineers, data scientists, and mathematicians.
Note: Matrix inverse (A⁻¹) is only defined when det(A) ≠ 0 (non-singular matrix). All determinant and inverse calculations use exact cofactor expansion and adjugate matrix methods.
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Matrix B
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Matrix Calculator — Multiply, Invert, Transpose, and Compute Determinants
Matrix operations are the computational backbone of linear algebra, which underlies machine learning, computer graphics, structural analysis, and systems of equations. Matrix multiplication is not commutative (AB ≠ BA in general) and requires the number of columns in the first matrix to equal the number of rows in the second. A 3×4 matrix times a 4×2 matrix gives a 3×2 result. The matrix calculator performs addition, subtraction, scalar multiplication, and matrix multiplication for matrices up to 5×5, showing the computation step by step.
The determinant of a square matrix measures "how much" the matrix transformation stretches or compresses space — a non-zero determinant means the matrix is invertible. A determinant of zero means the matrix is singular (not invertible) and the corresponding system of equations has either no solution or infinitely many. For a 2×2 matrix [a,b;c,d], det = ad - bc. For larger matrices, cofactor expansion or row reduction is used. The calculator computes determinants through row reduction for numerical stability, showing the elimination steps.
Matrix inversion solves systems of linear equations: if Ax = b, then x = A⁻¹b. The inverse exists only if det(A) ≠ 0. Gaussian elimination with back-substitution is computationally equivalent to computing the inverse and multiplying — both solve the linear system, but elimination is more numerically stable than explicitly forming A⁻¹. The calculator solves square systems up to 5×5 using elimination with partial pivoting, presenting both the inverse matrix (if requested) and the solution vector for a given right-hand side.