Inverse Matrix Calculator (2×2)
Calculate the inverse of a 2×2 matrix [[a, b], [c, d]], when it exists.
About the Inverse Matrix Calculator (2×2)
Not every matrix has an inverse — a 2×2 matrix [[a, b], [c, d]] is invertible only when its determinant, ad − bc, is not zero. When it is invertible, multiplying the matrix by its inverse gives the identity matrix [[1, 0], [0, 1]].
Formula: A−1 = (1 ÷ det(A)) × [[d, −b], [−c, a]], where det(A) = ad − bc
In other words: swap the two diagonal entries (a and d), negate the two off-diagonal entries (b and c), and divide every entry by the determinant. If the determinant is zero, the matrix is called singular and has no inverse — this calculator returns no result in that case, since dividing by a zero determinant is undefined.
Matrix inverses are used to solve systems of linear equations directly (x = A−1b), and they appear throughout computer graphics, economics and engineering wherever a linear transformation needs to be undone.