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Matrix Calculator

Enter a matrix and choose an operation to see step-by-step solutions. Supports addition, multiplication, determinants, inverses, and row reduction for matrices up to 5×5. All calculations run in your browser.

Matrix A×
A
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Reference Guide

Matrix Operations

Matrix addition and subtraction work element by element. Both matrices must have the same dimensions. Scalar multiplication multiplies every entry by the same number.

Addition
(A+B)ij=aij+bij(A + B)_{ij} = a_{ij} + b_{ij}
Scalar multiplication
(cA)ij=c⋅aij(cA)_{ij} = c \cdot a_{ij}

The transpose flips a matrix over its diagonal, swapping rows and columns so that (AT)ij=aji(A^T)_{ij} = a_{ji}.

Matrix Multiplication

Each entry of the product is the dot product of a row from the first matrix and a column from the second. The number of columns in A must equal the number of rows in B.

Product formula
(AB)ij=∑kaik bkj(AB)_{ij} = \sum_{k} a_{ik} \, b_{kj}
Dimension rule
Am×n⋅Bn×p=Cm×pA_{m \times n} \cdot B_{n \times p} = C_{m \times p}

Matrix multiplication is not commutative. In general, AB≠BAAB \ne BA even when both products are defined.

Determinant and Inverse

The determinant is a single number computed from a square matrix. For a 2×2 matrix it is ad−bcad - bc. Larger matrices use cofactor expansion along a row or column.

Invertibility condition
A−1 exists  ⟺  det⁡(A)≠0A^{-1} \text{ exists} \iff \det(A) \ne 0
Inverse identity
A⋅A−1=A−1⋅A=IA \cdot A^{-1} = A^{-1} \cdot A = I

When the determinant is zero, the matrix is singular and has no inverse. The rows (or columns) are linearly dependent.

Row Reduction

Row reduction uses three elementary operations: swap two rows, multiply a row by a nonzero scalar, and add a multiple of one row to another. These operations do not change the solution set of a system.

Row Echelon Form (REF) has all zeros below each leading entry (pivot). Reduced Row Echelon Form (RREF) goes further by making each pivot equal to 1 and clearing all entries above the pivots as well.

Applications Solving linear systems, finding matrix rank, and computing inverses
Rank
rank(A)=number of nonzero rows in REF\text{rank}(A) = \text{number of nonzero rows in REF}