Matrizenrechner
Matrizen addieren, subtrahieren und multiplizieren sowie Determinante, Inverse und Transponierte berechnen
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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Was ist eine Matrix?
Eine Matrix ist ein rechteckiges Zahlenschema, das in Zeilen und Spalten angeordnet ist. Matrizen bilden das Fundament der linearen Algebra und sind in Computergrafik, Ingenieurwesen, Wirtschaft, maschinellem Lernen und der Lösung linearer Gleichungssysteme allgegenwärtig. Ob ein 3D-Objekt gedreht, ein neuronales Netz trainiert oder ein Schaltkreis analysiert wird – Matrizen sind die mathematische Sprache dahinter.
Dieser Rechner unterstützt die gebräuchlichsten Operationen für 2×2- und 3×3-Matrizen: Addition, Subtraktion, skalare Multiplikation, Matrizenmultiplikation, Determinante, Inverse und Transponierte. Werte eingeben, Operation wählen und sofort das Ergebnis ablesen – ganz ohne Zettel und Stift.
So verwendest du den Matrizenrechner
- Wähle die gewünschte Operation aus dem Auswahlmenü.
- Gib die Matrizenwerte in das Eingaberaster ein (Zeile für Zeile, von links nach rechts).
- Bei zweimatrizigen Operationen (Addition, Subtraktion, Multiplikation) fülle sowohl Matrix A als auch Matrix B aus.
- Klicke auf Berechnen und lies das Ergebnis darunter ab.
Wichtige Formeln
Determinante 2×2:
|A| = ad − bc für A = [[a,b],[c,d]]
Inverse 2×2:
A⁻¹ = (1/|A|) × [[d,−b],[−c,a]]
Matrizenmultiplikation (A×B):
C[i][j] = Σ A[i][k] × B[k][j]
Transponierte: (Aᵀ)[i][j] = A[j][i]Die Matrizenmultiplikation ist NICHT kommutativ – A×B ist im Allgemeinen nicht gleich B×A. Eine Matrix besitzt keine Inverse, wenn ihre Determinante null ist (singuläre Matrix).
Durchgerechnete Beispiele
Determinante 2×2
Für A = [[3, 8], [4, 6]]: |A| = (3×6) − (8×4) = 18 − 32 = −14.
Matrizenaddition
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]: A + B = [[6, 8], [10, 12]].
Transponierte
Für A = [[1, 2, 3], [4, 5, 6]]: Aᵀ = [[1, 4], [2, 5], [3, 6]]. Zeilen werden zu Spalten.