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2×2 Matrix Inverse Calculator

Find the inverse of a 2×2 matrix.

How it is calculated

Enter the four entries of the matrix row by row: a and b in the top row, c and d in the bottom row. The tool first finds the determinant det = a·d − b·c. If it is exactly zero, the matrix is singular and the result shows singular with dashes instead of an inverse. Otherwise each entry of the inverse is a swapped or sign-changed entry divided by the determinant: A⁻¹ = (1 ÷ det) × [[d, −b], [−c, a]]. The determinant and the four inverse entries are displayed rounded to two decimals, so a value such as 1/3 appears as 0.33.

Worked example

Take the matrix with a = 4, b = 7, c = 2, d = 6. Determinant: 4 × 6 − 7 × 2 = 24 − 14 = 10.00. Divide the rearranged entries by 10: top row d ÷ det = 6 ÷ 10 = 0.60 and −b ÷ det = −7 ÷ 10 = −0.70; bottom row −c ÷ det = −2 ÷ 10 = −0.20 and a ÷ det = 4 ÷ 10 = 0.40. The inverse is [[0.60, −0.70], [−0.20, 0.40]]. Multiplying it by the original matrix gives the identity matrix, which is a good way to check the answer.

Try this example →

Common mistakes

  • Entering the entries column by column. The fields are a, b (first row) and c, d (second row); swapping b and c gives the inverse of the transpose instead.
  • Forgetting that rows with proportional entries, such as [2, 4] and [1, 2], have determinant 0 and no inverse. The tool reports singular for these.
  • Dividing each entry of the original matrix by the determinant. The inverse uses the rearranged entries d, −b, −c, a, not a, b, c, d.
  • Reusing the rounded two-decimal figures in later work. Entries like 1/3 are shown as 0.33, so multiplying back gives a result that is slightly off the identity matrix.
  • Expecting the inverse of a product to be in the same order. For two matrices, (AB)⁻¹ equals B⁻¹A⁻¹, not A⁻¹B⁻¹.

About this calculator

This calculator finds the inverse of a 2×2 matrix: for A = [[a,b],[c,d]], it computes the determinant (ad − bc) and then A⁻¹ = 1/det × [[d,−b],[−c,a]]. If the determinant is zero, the matrix has no inverse.

Students use it to check linear algebra homework, and it's the same math behind solving systems of two equations, computer graphics transformations (scaling, rotation, shearing), and cryptography exercises like the Hill cipher.

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Comments (1)

  • Editorial team

    The result is shown to two decimals, so a determinant like 0.004 appears as 0.00 even though the matrix is not singular and its inverse is made of very large entries. Treat a displayed determinant near zero as a warning that small rounding or measurement errors will be hugely amplified. To verify any answer, multiply it by your original matrix and look for the identity matrix. Are you inverting this for a system of equations or a transformation?

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