2x2 Matrix Inverse Calculator with Steps

Inverting a 2×2 matrix swaps the main diagonal, negates the off-diagonal, and divides by the determinant. For A = [[4,7],[2,6]]: det = 4×6 − 7×2 = 10, so A⁻¹ = (1/10)[[6,−7],[−2,4]] = [[0.6,−0.7],[−0.2,0.4]]. Multiplying back verifies: A·A⁻¹ = [[1,0],[0,1]] to full precision — the calculator performs this check on every run. Zero determinant aborts with a singular-matrix explanation (row dependence means no undo exists). The tool outputs fractions and decimals side by side (here 3/5 and −7/10), connects to systems solving: [[4,7],[2,6]]x = b solves as x = A⁻¹b, and notes the geometric effect — this inverse maps areas by 1/10 and undoes the original shear-and-scale. The determinant of the inverse equals 1/det(A) = 0.1.

1. 1

   Enter a, b, c, d

   The four entries of your matrix

2. 2

   Compute the determinant

   ad − bc = 10 here; zero flags singular

3. 3

   Swap and negate

   [[d,−b],[−c,a]] = [[6,−7],[−2,4]]

4. 4

   Divide and verify

   Scale by 1/det; A·A⁻¹ = I is confirmed

• 1

  Swap the diagonal, negate the wings, divide by det

• 2

  det = 0 ⇒ no inverse exists

• 3

  det(A⁻¹) = 1/det(A)

• Negating the diagonal instead of the off-diagonal

• Forgetting to divide by the determinant

• Applying the formula to singular matrices (det 0)

• Mixing entry order when typing the matrix