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The Inverse of a 2×2 Matrix

A matrix's inverse can be found directly from a formula, as long as its determinant is not zero.

Undoing a Matrix

Example 1 — finding an inverse

Find the inverse of A = (3254) . 1 Find the determinant (from an earlier lesson): det(A) = 2 2 Swap the main diagonal (3 and 4), negate the other diagonal (5 and 2): (4 - 2 - 5 3) 3 Divide every entry by the determinant (2): (4 \div 2 - 2 \div 2 - 5 \div 2 3 \div 2) Answer: A^{- 1} = (2 - 1 - 2.5 1.5)

Matrix lab

Matrix A

Determinant of A: 1.0000

A⁻¹ ≈ [[3,-1],[-5,2]]

Change a value and watch:

Change the matrix and watch the inverse update — notice how it depends on dividing by the determinant.

If you set the determinant to 0, the inverse formula breaks down — this is exactly the singular-matrix case from the last lesson.

Example 1 — a clean inverse

Find the inverse of A = (2513) . 1 det(A) = (2 \times 3) - (1 \times 5) = 6 - 5 = 1 2 Swap the main diagonal, negate the other: (3 - 5 - 1 2) 3 Divide every entry by 1 (which changes nothing here) Answer: A^{- 1} = (3 - 5 - 1 2)

Practice

Find the element in row 1, column 1 of the inverse of (4321) .

Your answer