Calculadora de Matrices
Suma, resta y multiplica matrices, y calcula el determinante, la inversa y la transpuesta
Matrix Calculator
Calculate the determinant of a 2x2 matrix
Enter matrix values [a b; c d]
det = ad - bcWhat is a Matrix Calculator?
A Matrix Calculator is a math tool for working with matrices—rectangular grids of numbers arranged in rows and columns. Matrices are used to represent and solve problems involving systems of equations, transformations, and data organized in table form. They are foundational in algebra, calculus, statistics, engineering, physics, computer graphics, machine learning, and many other fields.
Common matrix operations include addition, subtraction, multiplication, transpose, determinant, and inverse. Doing these operations by hand can be time-consuming and easy to get wrong, especially for 3×3 or larger matrices. A matrix calculator helps you compute results instantly and accurately.
Common Matrix Operations
- Add (A + B) -- add corresponding entries
- Subtract (A − B) -- subtract corresponding entries
- Multiply (A × B) -- row-by-column dot products
- Transpose (Aᵀ) -- flip rows and columns
- Determinant (det(A)) -- a scalar that describes matrix properties
- Inverse (A⁻¹) -- the matrix that undoes A
Matrices are especially important for solving multiple linear equations at once, transforming coordinates in 2D/3D graphics (rotation, scaling), modeling networks and relationships (graphs, Markov chains), and representing datasets and computations in engineering and science.
How to Use This Matrix Calculator
- Choose the matrix size -- for example 2×2, 3×3, etc., if the calculator supports sizing
- Enter the matrix values -- fill in the grid (Matrix A, and Matrix B if needed)
- Select the operation -- such as add, subtract, multiply, transpose, determinant, or inverse
- Click "Calculate" -- to generate the result
- Review the output -- confirm the dimensions match what you expect
Tips:
- Addition / Subtraction: matrices must be the same size
- Multiplication: the number of columns in A must equal the number of rows in B
- Inverse: only square matrices (like 2×2, 3×3) can have an inverse, and only if the determinant is not zero
Matrix Formulas
Addition / Subtraction
For matrices A and B of the same size:
(A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ
Add or subtract corresponding entries
Transpose
Flip rows and columns:
(Aᵀ)ᵢⱼ = Aⱼᵢ
Row i becomes column i
Matrix Multiplication
If A is m×n and B is n×p, then A×B is m×p:
(AB)ᵢⱼ = Σ Aᵢₖ × Bₖⱼ (k = 1 to n)
Each entry is a dot product of a row of A and a column of B
Determinant (2×2)
For A = [a b; c d]:
det(A) = ad − bc
Larger matrices use cofactor expansion
Inverse (2×2)
If det(A) ≠ 0:
A⁻¹ = (1/det(A)) × [d, −b; −c, a]
3×3+ uses cofactors or row reduction
Example Calculations
Example 1: Matrix Addition
A = [1 2; 3 4], B = [5 6; 7 8]
Calculation: add entry-by-entry
Result: [6 8; 10 12]
Example 2: Matrix Multiplication
A = [1 2; 3 4], B = [2 0; 1 2]
Calculation:
- (1,1): 1×2 + 2×1 = 4
- (1,2): 1×0 + 2×2 = 4
- (2,1): 3×2 + 4×1 = 10
- (2,2): 3×0 + 4×2 = 8
Result: [4 4; 10 8]
Example 3: Transpose
A = [1 2 3; 4 5 6] (2×3 matrix)
Transpose: flip rows and columns
Result: Aᵀ = [1 4; 2 5; 3 6] (3×2 matrix)
Example 4: Determinant and Inverse (2×2)
A = [4 7; 2 6]
Determinant: 4×6 − 7×2 = 24 − 14 = 10
Since det(A) ≠ 0, inverse exists:
A⁻¹ = (1/10) × [6, −7; −2, 4] = [0.6, −0.7; −0.2, 0.4]
Frequently Asked Questions
What is a matrix used for?
Matrices represent structured data and transformations. They're used to solve systems of linear equations, perform coordinate transformations in graphics, model networks, and handle computations in science and engineering.
When can I add or subtract matrices?
Only when they have the same dimensions (same number of rows and columns). You add or subtract corresponding entries.
When can I multiply matrices?
Matrix multiplication requires that the number of columns in A equals the number of rows in B. If A is m×n, B must be n×p.
What does the determinant tell me?
The determinant is a single number that indicates properties of a square matrix. If det(A) = 0, the matrix is singular (not invertible). If det(A) is not zero, an inverse exists.
Why doesn't my matrix have an inverse?
A matrix must be square and have a non-zero determinant to be invertible. If the determinant is 0, the matrix has no inverse (it is called singular).
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¿Qué es una matriz?
Una matriz es un arreglo rectangular de números dispuestos en filas y columnas. Las matrices son la base del álgebra lineal y están presentes en gráficos por computadora, ingeniería, economía, aprendizaje automático y en la resolución de sistemas de ecuaciones. Desde rotar un objeto 3D hasta entrenar una red neuronal, las matrices son el lenguaje matemático detrás de todo.
Esta calculadora resuelve las operaciones más comunes con matrices de 2×2 y 3×3: suma, resta, multiplicación por escalar, multiplicación de matrices, determinante, inversa y transpuesta. Solo ingresa tus valores, elige la operación y obtén el resultado al instante, sin necesidad de papel ni cálculo manual.
Cómo usar la calculadora de matrices
- Selecciona la operación que deseas realizar en el menú desplegable.
- Ingresa los valores de la matriz en la cuadrícula (fila por fila, de izquierda a derecha).
- Para operaciones con dos matrices (suma, resta, multiplicación), completa tanto la Matriz A como la Matriz B.
- Haz clic en Calcular y lee el resultado debajo.
Fórmulas clave
Determinante 2×2:
|A| = ad − bc para A = [[a,b],[c,d]]
Inversa 2×2:
A⁻¹ = (1/|A|) × [[d,−b],[−c,a]]
Multiplicación de matrices (A×B):
C[i][j] = Σ A[i][k] × B[k][j]
Transpuesta: (Aᵀ)[i][j] = A[j][i]La multiplicación de matrices NO es conmutativa: en general A×B es distinto de B×A. Una matriz no tiene inversa cuando su determinante es 0 (se llama matriz singular).
Ejemplos resueltos
Determinante 2×2
Para A = [[3, 8], [4, 6]]: |A| = (3×6) − (8×4) = 18 − 32 = −14.
Suma de matrices
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]: A + B = [[6, 8], [10, 12]].
Transpuesta
Para A = [[1, 2, 3], [4, 5, 6]]: Aᵀ = [[1, 4], [2, 5], [3, 6]]. Las filas se convierten en columnas.