Matrix Calculator

Perform matrix operations including addition, subtraction, multiplication, transpose, determinant, and inverse calculations.

Result

Result Matrix

2×2

See below for values

OperationAdd
Matrix A Size2×2
Matrix B Size2×2

Select Operation

Matrix A

Matrix A (2×2)

Matrix B

Matrix B (2×2)

Result

6.0000
8.0000
10.0000
12.0000

Matrix Operation Rules

Addition / Subtraction

Same dimensions required. Add/subtract corresponding elements.

Multiplication

A(m×n) × B(n×p) = C(m×p). Not commutative: AB ≠ BA.

Transpose

Aᵀ swaps rows and columns. (AB)ᵀ = BᵀAᵀ.

Determinant

Square matrices only. det(AB) = det(A)×det(B).

Inverse

Square matrix with det ≠ 0. A×A⁻¹ = I.

Scalar Multiply

Multiply every element by the scalar k.

Special Matrices

Identity Matrix (I)

1s on diagonal, 0s elsewhere

[1 0]
[0 1]

Zero Matrix

All elements are 0

[0 0]
[0 0]

Diagonal Matrix

Non-zero only on diagonal

[a 0]
[0 b]

Result

Result Matrix

2×2

See below for values

OperationAdd
Matrix A Size2×2
Matrix B Size2×2

?How Do You Calculate Matrices?

A matrix is a rectangular array of numbers. Operations: Addition/subtraction (element-wise, same size). Multiplication (rows by columns, inner dimensions must match). Determinant (scalar value, square matrices). Inverse (A^-1 where A*A^-1 = I). Transpose (swap rows and columns). Used in linear algebra, graphics, and systems of equations.

What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrix algebra provides operations for manipulating these arrays, fundamental to linear algebra, computer graphics, physics, engineering, and data science. Matrix operations solve systems of equations, perform transformations, and model linear relationships.

Key Facts About Matrices

  • Matrix dimensions: rows x columns (m x n)
  • Addition/subtraction: element-wise, same dimensions required
  • Multiplication: (m x n) * (n x p) = (m x p)
  • Determinant: scalar value, only for square matrices
  • Inverse exists only if determinant is not equal to 0
  • Transpose: swap rows and columns (A^T)
  • Identity matrix (I): diagonal 1s, 0s elsewhere
  • Used in solving systems of linear equations: Ax = b

Quick Answer

A matrix is a rectangular array of numbers. Operations: Addition/subtraction (element-wise, same size). Multiplication (rows by columns, inner dimensions must match). Determinant (scalar value, square matrices). Inverse (A^-1 where A*A^-1 = I). Transpose (swap rows and columns). Used in linear algebra, graphics, and systems of equations.

Frequently Asked Questions

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used in linear algebra, computer graphics, statistics, and machine learning. A matrix with m rows and n columns is called an m×n matrix.
To multiply A×B, the number of columns in A must equal rows in B. Each element (i,j) in the result equals the dot product of row i from A and column j from B. The result has dimensions (rows of A) × (columns of B).
The determinant is a scalar value computed from a square matrix. It indicates if the matrix is invertible (det ≠ 0) and represents the scaling factor for area/volume when the matrix is used as a transformation. For 2×2: det = ad - bc.
The inverse of matrix A (written A⁻¹) is the matrix where A × A⁻¹ = I (identity matrix). Only square matrices with non-zero determinant have inverses. For 2×2: swap diagonal, negate off-diagonal, divide by determinant.
The transpose of a matrix switches rows and columns. Element at (i,j) moves to (j,i). An m×n matrix becomes n×m. The transpose of Aᵀᵀ equals A. For symmetric matrices, A = Aᵀ.

Last updated: 2025-01-15