Last updated: 2026-07-01

What this calculator does

This calculator finds the inverse of a 2×2 matrix. Enter the four entries of A = [[a, b], [c, d]] and it returns the determinant and the four entries of A⁻¹, along with a step-by-step derivation. When the matrix has no inverse, it says so instead of printing meaningless numbers.

The inverse of a matrix A is written $A^{-1}$, and it is defined by the relation $A A^{-1} = A^{-1} A = I$, where $I$ is the identity matrix.

The 2×2 inverse formula

For a 2×2 matrix, the inverse comes from the adjugate divided by the determinant:

There is an easy way to remember the matrix part: swap the two main-diagonal entries ($a$ and $d$), negate the two anti-diagonal entries ($b$ and $c$), then divide everything by the determinant $ad - bc$.

The determinant comes first

The determinant $\det A = ad - bc$ decides whether an inverse exists at all. If it is non-zero, the formula above gives the unique inverse. If it is zero, the matrix is singular and has no inverse — there is no matrix you can multiply by to get back the identity.

Worked example

Take the matrix [[4, 3], [6, 3]].

First the determinant:

$\det A = (4)(3) - (3)(6) = 12 - 18 = -6$

Since the determinant is not zero, the inverse exists. Apply the formula:

Checking the answer

A good habit is to multiply the original matrix by the inverse and confirm you get the identity:

The product is the identity matrix, which confirms the inverse is correct. Tiny differences in the last decimal place come only from rounding the displayed entries.

When the inverse does not exist

A 2×2 matrix is singular when $ad - bc = 0$. Geometrically, its two column vectors lie on the same line, so the transformation flattens the plane onto a line and information is lost — there is no way to reverse it. For example, [[2, 4], [1, 2]] has determinant $(2)(2) - (4)(1) = 0$, so it has no inverse.

Solving Ax = b with the inverse

One of the main uses of the inverse is solving a linear system. If $A\mathbf{x} = \mathbf{b}$ and $A$ is invertible, then multiplying both sides by $A^{-1}$ gives

$\mathbf{x} = A^{-1}\mathbf{b}.$

For a 2×2 system this is a quick route to the solution once the inverse is known. When the coefficient matrix is singular, the system instead has either no solution or infinitely many.

Useful properties

• $(A^{-1})^{-1} = A$ — inverting twice returns the original matrix.
• $(AB)^{-1} = B^{-1} A^{-1}$ — the inverse of a product reverses the order.
• $\det(A^{-1}) = 1 / \det(A)$ — the determinant of the inverse is the reciprocal of the determinant.
• $(A^{\mathsf{T}})^{-1} = (A^{-1})^{\mathsf{T}}$ — transpose and inverse commute.