2x2 Matrix Calculator

A 2x2 matrix is a grid of four numbers arranged in two rows and two columns. Matrix operations are fundamental in linear algebra and are used in computer graphics, physics simulations, engineering, and data science.

For matrix A = [[a, b], [c, d]] and matrix B = [[e, f], [g, h]], this calculator supports five operations:

Addition: Add corresponding elements. A + B = [[a+e, b+f], [c+g, d+h]]

Subtraction: Subtract corresponding elements. A - B = [[a-e, b-f], [c-g, d-h]]

Multiplication: Each element in the result is the dot product of a row from A and a column from B. Note that matrix multiplication is not commutative (A × B does not always equal B × A). A × B = [[ae+bg, af+bh], [ce+dg, cf+dh]]

Determinant: A single number that tells you whether the matrix is invertible and how it scales area. det(A) = ad - bc

Inverse: The matrix that, when multiplied by A, gives the identity matrix. It only exists when the determinant is not zero. A⁻¹ = (1/det(A)) × [[d, -b], [-c, a]]

Practical example: For A = [[3, 1], [2, 4]], the determinant is (3×4) - (1×2) = 10. Since the determinant is not zero, the inverse exists: A⁻¹ = (1/10) × [[4, -1], [-2, 3]] = [[0.4, -0.1], [-0.2, 0.3]].

Tips: If the determinant is zero, the matrix is called “singular” and has no inverse. The determinant also tells you the scaling factor: a determinant of 2 means the matrix doubles areas, while a negative determinant means it also reflects (flips) the space.