Last updated: April 2026 · Reviewed by Calculatormatics Editorial Team

Matrix Calculator — Addition, Multiplication, Transpose, Determinant & Inverse (2×2 to 4×4)

A matrix is a rectangular array of numbers arranged in rows and columns. Matrix algebra underpins computer graphics (3D rotation and scaling), physics (quantum mechanics, stress tensors), engineering (finite element analysis), and machine learning (neural networks, PCA, linear regression). This calculator handles the six most common matrix operations: addition, subtraction, multiplication, transpose, determinant, and inverse — for square matrices from 2×2 up to 4×4. Enter your values, choose an operation, and the result appears instantly with calculation steps shown below. Unlike most online tools, this calculator shows the step-by-step arithmetic so you can follow and verify each computation yourself.

Matrix multiplication: C[i][j] = sum over k of A[i][k] × B[k][j]

C[1][1] = 1×5 + 2×7 = 19
C[1][2] = 1×6 + 2×8 = 22
C[2][1] = 3×5 + 4×7 = 43
C[2][2] = 3×6 + 4×8 = 50

Matrix Basics

A matrix is described by its dimensions: m rows × n columns. An individual element is written as a_ij, where i is the row index and j is the column index (both starting at 1). For example, in a 3×3 matrix A, the element in row 2 column 3 is a₂₃.

Not all operations are defined for all pairs of matrices. The rules are:

• Addition / Subtraction (A ± B): Both matrices must have the same shape (same number of rows and same number of columns).
• Multiplication (A × B): The number of columns in A must equal the number of rows in B. If A is m×n and B is n×p, the result is m×p.
• Transpose (A^T): Defined for any matrix. An m×n matrix becomes n×m.
• Determinant and Inverse: Only defined for square matrices (same number of rows and columns).

Because this calculator uses the same size for A and B, addition, subtraction, and multiplication are always dimension-compatible.

Matrix Multiplication

Matrix multiplication is the most conceptually demanding of the six operations. The element C[i][j] of the result is computed as the dot product of row i of A with column j of B:

c_ij = sum over k of (a_ik × b_kj)
     = a_i1×b_1j + a_i2×b_2j + ... + a_in×b_nj

A key property: matrix multiplication is not commutative. That is, A × B and B × A generally produce different results (and sometimes one is defined while the other is not). This is fundamentally different from multiplying ordinary numbers, where 3 × 5 = 5 × 3 always. In matrix algebra, the order matters.

Matrix multiplication is, however, associative: (A × B) × C = A × (B × C). And it is distributive over addition: A × (B + C) = A×B + A×C.

Worked Example: 2×2 Multiplication

Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. We compute each element of C = A × B:

A = | 1  2 |    B = | 5  6 |
    | 3  4 |        | 7  8 |

c[1][1] = 1×5 + 2×7 = 5 + 14  = 19
c[1][2] = 1×6 + 2×8 = 6 + 16  = 22
c[2][1] = 3×5 + 4×7 = 15 + 28 = 43
c[2][2] = 3×6 + 4×8 = 18 + 32 = 50

Result C = A × B = | 19  22 |
                   | 43  50 |

Notice that B × A would give a different result (check: B×A[1][1] = 5×1 + 6×3 = 23 ≠ 19), confirming that matrix multiplication is not commutative.

The Transpose

The transpose of a matrix A, written A^T, is formed by swapping rows and columns. Formally: A^T[i][j] = A[j][i]. An m×n matrix becomes n×m after transposition.

If A = | 1  2  3 |
       | 4  5  6 |

Then A^T = | 1  4 |
           | 2  5 |
           | 3  6 |

The transpose has important special cases:

• Symmetric matrix: A = A^T. The matrix is its own transpose. Covariance matrices in statistics are always symmetric.
• Orthogonal matrix: A^T = A^-1. The transpose equals the inverse, so A × A^T = I. Rotation matrices in 3D graphics are orthogonal.
• (AB)^T = B^TA^T: The transpose reverses the order of multiplication.

The Determinant

The determinant is a single number derived from a square matrix that encodes geometric and algebraic information about the transformation the matrix represents. Geometrically, |det(A)| is the scale factor by which the transformation changes volumes (or areas in 2D). If det(A) is negative, the transformation also reverses orientation (like a reflection).

For a 2×2 matrix:

A = | a  b |
    | c  d |

det(A) = a×d − b×c

For a 3×3 matrix, the determinant is computed by cofactor expansion along the first row:

det(A) = a₁₁×(a₂₂×a₃₃ − a₂₃×a₃₂)
       − a₁₂×(a₂₁×a₃₃ − a₂₃×a₃₁)
       + a₁₃×(a₂₁×a₃₂ − a₂₂×a₃₁)

When det(A) = 0, the matrix is singular: it collapses space (one or more rows are linearly dependent), and no inverse exists. When det(A) = 1, the transformation preserves volume exactly. Rotation matrices always have det = 1; reflection matrices have det = -1.

The Inverse

The inverse of a square matrix A, written A^-1, is the matrix that "undoes" the transformation: A × A^-1 = A^-1 × A = I (the identity matrix). An inverse exists if and only if det(A) ≠ 0 (the matrix is non-singular).

For a 2×2 matrix, the inverse is given by the adjugate formula:

A = | a  b |          A⁻¹ = (1/det) × |  d  −b |
    | c  d |                           | −c   a |

where det = a×d − b×c

For 3×3 and 4×4 matrices, the process involves computing the cofactor matrix (replacing each element with the signed determinant of the submatrix obtained by deleting its row and column), transposing it to form the adjugate, and dividing by the determinant. This calculator uses the mathjs library, which implements numerically stable algorithms for all sizes up to 4×4.

The inverse is fundamental to solving linear systems: if Ax = b (where b is a vector of known values and x is the unknown), then x = A^-1b. This is used everywhere from electrical circuit analysis to structural engineering to machine learning regression.

Applications

Matrix operations appear in a surprisingly wide range of fields:

• Computer Graphics: Every rotation, scaling, and translation of a 3D object is encoded as a 4×4 matrix. When you move a character in a video game, the GPU multiplies millions of vertex coordinates by transformation matrices per frame. Combining multiple transformations is just matrix multiplication.
• Solving Linear Systems (Ax = b): Any system of n linear equations in n unknowns can be written in matrix form. Finding the solution is equivalent to computing x = A^-1b, used in electrical engineering (Kirchhoff's laws), structural analysis, and economic modelling.
• Principal Component Analysis (PCA) in ML: PCA finds the directions of maximum variance in a dataset by computing the eigenvectors of the covariance matrix (a symmetric matrix). This requires computing determinants, transposes, and inverses — all on this calculator.
• Physics — Moment of Inertia Tensor: The rotational inertia of a 3D rigid body is a 3×3 symmetric matrix. Diagonalising it (finding the principal axes) requires computing its eigenvectors.
• Statistics — Multivariate Normal Distribution: The probability density involves the inverse and determinant of the covariance matrix. Every Gaussian model in machine learning uses this.

Frequently Asked Questions