2×2 Determinant Calculator

  Formula for the determinant of a 2×2 matrix

A 2×2 matrix has two rows and two columns. Call the entries of the first row a and b, and the entries of the second row c and d. The determinant is the product of the main diagonal minus the product of the other diagonal:

det(A) = a·d − b·c

That single subtraction is the whole calculation. No expansion and no row reduction are needed, which is why the 2×2 case is the building block for every larger determinant: cofactor expansion of a 3×3 matrix ends in three 2×2 determinants.

  Worked example

Take the matrix with rows (3, 5) and (2, 4). The main diagonal gives 3·4 = 12. The other diagonal gives 5·2 = 10. So det(A) = 12 − 10 = 2.

Watch the signs when entries are negative. For rows (−1, 6) and (2, −3) the main diagonal gives (−1)·(−3) = 3 and the other diagonal gives 6·2 = 12, so the determinant is 3 − 12 = −9.

  What the result tells you

If the determinant is zero, the two rows are multiples of each other, the matrix is singular and it has no inverse. For rows (2, 4) and (1, 2) you get 2·2 − 4·1 = 0. If the determinant is not zero, the matrix is invertible and the same number appears in the denominator of the 2×2 inverse formula.

Geometrically, the absolute value of the determinant is the area of the parallelogram spanned by the two rows, and the sign says whether the orientation is preserved or flipped.

The calculator above is set to a 2×2 matrix. Enter integers, decimals, fractions or complex numbers and it shows the same two products and the subtraction.