The determinant of a 2x2 matrix [ [a, b], [c, d] ] is:

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For a 2x2 matrix, the determinant is the difference between the product of the entries on one diagonal and the product on the other diagonal. This comes from the rule for 2x2 determinants and also reflects how the linear transformation scales areas.

So for the matrix [ [a, b], [c, d] ], the determinant is ad − bc. It’s the precise combination that measures how the area is stretched or compressed by the transformation.

To see why this works, think of expanding along the first row: a times the determinant of [ [d] ] minus b times the determinant of [ [c] ], which gives ad − bc.

Check with a quick example: a = 2, b = 3, c = 4, d = 5 yields 2×5 − 3×4 = 10 − 12 = −2.

The other expressions don’t match the determinant: ab − cd mixes the terms in a way that doesn’t correspond to the area-scaling rule; a + d − b − c is just a different algebraic combination; ad + bc is the sum of the diagonal products, not their difference.