Two by Two Determinant Formula

Quick answer: For A = [[a,b],[c,d]], det(A) = ad − bc. Subtract the product of the other diagonal from the product of the main diagonal.

Worked example

Find the determinant of [[3,2],[5,4]].

1. Main diagonal product: 3×4 = 12.
2. Other diagonal product: 2×5 = 10.
3. Determinant = 12 − 10 = 2.

Check the result

Swapping the two rows gives determinant 5×2 − 4×3 = −2, the negative of the original.

Common mistake to avoid

The determinant is a number, not another 2×2 matrix. Preserve signs for negative entries; do not add both diagonal products.

Try it yourself

1. Find det([[1,2],[2,4]]). 2. Find det([[−1,3],[2,4]]).

Answers and reasoning

1. 4 − 4 = 0; rows are proportional. 2. (−1)4 − 3×2 = −10.

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