▦ Matrix Calculator
Perform 2×2 matrix operations including addition, subtraction, multiplication, determinant, and transpose. Enter values for Matrix A (and Matrix B where needed).
Enter values for Matrix A and Matrix B, then choose Add or Subtract.
Matrix A
Matrix B
Result
Matrix Formulas
Worked Examples
Example — Multiplication
A = [[1,2],[3,4]], B = [[5,6],[7,8]]
C[0][1] = 1×6 + 2×8 = 22
C[1][0] = 3×5 + 4×7 = 43
C[1][1] = 3×6 + 4×8 = 50
Example — Determinant
A = [[3,8],[4,6]]
Frequently Asked Questions
Matrix multiplication combines two matrices by computing dot products of rows and columns. For 2×2 matrices A and B, element C[i][j] = A[i][0]×B[0][j] + A[i][1]×B[1][j]. Unlike numbers, matrix multiplication is NOT commutative — A×B ≠ B×A in general.
The determinant is a scalar value computed from a square matrix. For a 2×2 matrix [[a,b],[c,d]], det = ad−bc. It encodes geometric information: the absolute value equals the area scaling factor of the linear transformation represented by the matrix.
A square matrix is invertible (non-singular) if and only if its determinant is non-zero. If det(A) = 0, the matrix is singular and has no inverse, meaning the corresponding system of equations has either no solution or infinitely many solutions.
The transpose Aᵀ is formed by flipping the matrix over its main diagonal — swapping rows and columns. If A is m×n, then Aᵀ is n×m. A key property: (A×B)ᵀ = Bᵀ×Aᵀ.
Related Calculators
Algebra Calculator
Solve algebraic expressions and evaluate equations step by step.
Equation Solver
Solve linear and quadratic equations for x step by step.
Scientific Calculator
Full-featured scientific calculator with trigonometry, logarithms, and advanced math functions.