Perform matrix operations on 2×2 and 3×3 matrices. Calculate determinant, inverse, transpose, addition, subtraction, and multiplication with step-by-step working shown.
2×2 determinant: det(A) = ad − bc for [[a,b],[c,d]]
2×2 inverse: A⁻¹ = (1/det(A)) × [[d,−b],[−c,a]]
Dot product: a·b = Σ(aᵢ × bᵢ) (scalar result)
Cross product: a×b = [a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁] (3D vector result)
For a 2×2 matrix [[a,b],[c,d]], the determinant = ad − bc.
The inverse of [[a,b],[c,d]] = (1/det) × [[d,−b],[−c,a]], where det = ad − bc. The inverse only exists if det ≠ 0.
A zero determinant means the matrix is singular (non-invertible) — the system of equations it represents has no unique solution.
To multiply matrix A (m×n) by matrix B (n×p), each element [i][j] of the result = the dot product of row i of A and column j of B. The number of columns in A must equal the number of rows in B.
The transpose flips a matrix over its diagonal — rows become columns and columns become rows. Element [i][j] becomes [j][i].