2x2 Matrix Determinant Calculator

The 2×2 determinant is one multiplication cross: ad − bc. For the matrix [[3,1],[2,4]] it is 3×4 − 1×2 = 10. The sign carries geometry — the determinant measures signed area of the parallelogram the matrix’s columns span, so [[3,1],[2,4]] scales area by exactly 10, and a negative determinant means orientation flipped. Zero is the critical signal: det = 0 means the matrix is singular, has no inverse, and collapses the plane onto a line. The calculator prints the two products (12 and 2), their difference, the singularity verdict, and — when invertible — the inverse matrix (1/det)×[[d,−b],[−c,a]], here [[0.4,−0.1],[−0.2,0.3]]. Eigenvalue support follows: the characteristic polynomial λ² − (trace)λ + det = 0, so this matrix has λ² −7λ +10 = 0 and eigenvalues 5 and 2.

1. 1

   Enter the four entries

   a, b on the first row; c, d on the second

2. 2

   Cross-multiply

   Compute a·d and b·c — both products are shown

3. 3

   Subtract

   det = ad − bc = 10 for the example

4. 4

   Interpret the result

   Area scale, invertibility, and the inverse when det ≠ 0

• 1

  det = ad − bc — the main diagonal minus the anti-diagonal

• 2

  Zero determinant = no inverse exists

• 3

  Eigenvalues of 2×2 solve λ² − trace·λ + det = 0

• Computing bc − ad, flipping the determinant’s sign

• Assuming a determinant of 0 still has an inverse

• Adding the products instead of subtracting (10 vs 14 here)

• Mixing row order when entering the matrix