Matrix Operations Calculator: Linear Algebra

Matrix Operations in Linear Algebra

Matrices are rectangular arrays of numbers used to represent linear transformations, systems of equations, and data. Matrix operations — multiplication, inversion, determinants — form the backbone of linear algebra and are fundamental to computer graphics, machine learning, physics simulations, and engineering analysis.

📊 Matrix Operations Calculator

Free calculator for instant results.

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📐 Formula

(AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ

Matrix multiplication: element (i,j) of product = dot product of row i of A with column j of B. A is m×n, B is n×p → AB is m×p.

📝 Worked Example

A=[[1,2],[3,4]], B=[[5,6],[7,8]]:
AB[0,0] = 1×5+2×7 = 19
AB[0,1] = 1×6+2×8 = 22
AB[1,0] = 3×5+4×7 = 43 → AB = [[19,22],[43,50]]

📝 How to Use

Enter Matrix AInput rows separated by newlines, values by spaces or commas.

Select OperationAdd, subtract, multiply, transpose, inverse, determinant, rank.

Enter Matrix BIf required for the selected operation.

CalculateView the result matrix with step-by-step working.

❓ FAQ

When is matrix multiplication not commutative?

Almost always: AB ≠ BA. Multiplication is only commutative for identity matrices, diagonal matrices, and their inverses. Order matters!

What does the determinant represent geometrically?

For a 2×2 matrix, |det| = area scaling factor of the transformation. det=0: matrix is singular (collapses space), no inverse exists.

Matrix Operations in Linear Algebra