◆ LINEAR ALGEBRA
Matrix Calculator
Add, subtract, and multiply matrices. Find determinant, inverse, transpose, scalar multiples, and matrix properties. Supports up to 5×5 with step-by-step solutions.
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OPERATION
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RESULT SIZE
Matrix A
Single Matrix Operations
Result
Size: 3×31
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Step-by-step
Rows become columns: result[j][i] = A[i][j]. A is 3×3 → Aᵀ is 3×3.
Understanding Matrix Operations
A matrix is simply a rectangular grid of numbers arranged in rows and columns — a compact way to store and transform structured data. Matrix operations form the backbone of linear algebra, the branch of math that powers 3D computer graphics and game engines, machine learning models, physics and engineering simulations, and even how search engines rank pages across a network.
Key Matrix Formulas
Addition (A+B) / Subtraction (A−B): only defined when both matrices share the same dimensions — combine matching cells one by one.
Multiplication (A×B): A must be m×n and B must be n×p (inner dimensions match) → result is m×p, where C[i][j] = Σₖ A[i][k] × B[k][j].
Transpose (Aᵀ): flip rows into columns, so an m×n matrix becomes n×m — result[j][i] = A[i][j].
Determinant, 2×2: det = ad − bc.
Determinant, 3×3 and larger: expand by cofactors along the first row, recursing down to 2×2 minors.
Inverse (A⁻¹): only exists when det(A) ≠ 0, computed as A⁻¹ = adjugate(A) ÷ det(A).
Trace: the sum of the elements on the main diagonal.
Hadamard product (A⊙B): same-size matrices multiplied cell by matching cell.— FAQ