Determinants and inverses

The determinant

For a 2×2 matrix [[a, b], [c, d]] the determinant is

det = a·d − b·c

It has a picture: the matrix stretches the unit square into a parallelogram, and |det| is that parallelogram's area. A negative determinant means the shape was also flipped over, like a mirror image.

When det is 0

Determinant 0 means the plane is squashed flat onto a line (or a point). Information is lost: different points land on the same place, so the transformation cannot be undone.

The inverse

The inverse A⁻¹ undoes A: A·A⁻¹ = I. It exists exactly when det is not 0. For a 2×2 matrix:

A⁻¹ = (1 ÷ det) × [[d, −b], [−c, a]]

Swap a and d, change the signs of b and c, and divide by the determinant.

from fractions import Fraction

def det2(m):
    return m[0][0] * m[1][1] - m[0][1] * m[1][0]

def inverse2(m):
    d = det2(m)
    if d == 0:
        return None
    return [[Fraction(m[1][1], d), Fraction(-m[0][1], d)],
            [Fraction(-m[1][0], d), Fraction(m[0][0], d)]]

A = [[4, 7], [2, 6]]
print("det A =", det2(A))
for row in inverse2(A):
    print("inverse row:", " ".join(str(x) for x in row))
flat = [[1, 2], [2, 4]]
print("det of [[1, 2], [2, 4]] =", det2(flat), "so its inverse is", inverse2(flat))

Bigger matrices

For larger matrices, run Gaussian elimination: the determinant is the product of the pivots, with a sign change for every row swap. Solving A·x = b this way is also faster and more accurate than computing A⁻¹ first.

Common mistakes

• Computing a·d + b·c. The determinant subtracts: a·d − b·c.
• Expecting every matrix to have an inverse. [[1, 2], [2, 4]] has det 4 − 4 = 0: its second row is twice the first.

Mark lesson complete