Determinant (2×2)
Determinant of a 2×2 matrix.
Formula
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Example
det[[1,2],[3,4]] = 1·4 − 2·3 = −2.
How it works
The determinant of a 2×2 matrix gives a single number that measures how the matrix transforms area and orientation in the plane. For the matrix [[a, b], [c, d]], the quantity ad − bc comes from comparing the two diagonal products: one contributes positively and the other negatively because the determinant must change sign when two rows or columns are swapped. If the determinant is 0, the transformation collapses the plane onto a line, so the matrix is not invertible.
When to use it
Use this formula when you need to test whether a 2×2 matrix is invertible, solve small linear systems, or compute the area-scaling factor of a 2D linear transformation. It also appears in geometry, physics, and coordinate transformations whenever 2D vectors or motions are involved.
Worked example, step by step
For the matrix [[5, -1], [2, 3]]: Step 1: Identify the entries: a = 5, b = -1, c = 2, d = 3. Step 2: Compute the main diagonal product: ad = 5 × 3 = 15. Step 3: Compute the other diagonal product: bc = (-1) × 2 = -2. Step 4: Subtract: ad − bc = 15 − (-2) = 17. Final answer: det([[5, -1], [2, 3]]) = 17.
Common mistakes to avoid
- Adding the diagonal products instead of subtracting: ad + bc
- Mixing up the positions of b and c when multiplying
- Forgetting that a negative bc becomes subtraction of a negative number, which turns into addition
Did You Know?
A determinant of zero means the matrix squashes space flat — and has no inverse.
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