Inverse of a Two by Two Matrix

Quick answer: For A = [[a,b],[c,d]] with ad − bc ≠ 0, A⁻¹ = (1/(ad − bc))[[d,−b],[−c,a]]. A zero determinant means no inverse exists.

Worked example

Find the inverse of A = [[2,1],[5,3]].

1. det(A) = 2×3 − 1×5 = 1.
2. Swap diagonal entries 2 and 3.
3. Negate off-diagonal entries 1 and 5.
4. Divide by determinant 1: A⁻¹ = [[3,−1],[−5,2]].

Check the result

AA⁻¹ = [[6−5,−2+2],[15−15,−5+6]] = [[1,0],[0,1]].

Common mistake to avoid

Do not replace each entry by its reciprocal. Always check determinant first to avoid division by zero.

Try it yourself

1. Invert [[1,0],[0,2]]. 2. Can [[1,2],[2,4]] be inverted?

Answers and reasoning

1. [[1,0],[0,1/2]]. 2. No: determinant is 4 − 4 = 0.

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