The 2×2 Determinant

Linear Algebra (Matrices) · Axiom Academy

One formula does the work: — the main-diagonal product minus the anti-diagonal product. 1. The Rule: Diagonal Minus Anti-Diagonal Every matrix has a single number attached to it called the determinant . To get it, multiply the two entries on the main diagonal (top-left × bottom-right), multiply the two on the anti-diagonal (top-right × bottom-left), and subtract the second product from the first. ad — product down the main diagonal bc — product down the anti-diagonal (subtracted) 2. Computing It on a Real Matrix Watch the rule run on a concrete matrix. The main diagonal lights up, then the anti-diagonal, then the two products meet at the subtraction. 3. What That Number Means: Signed Area The columns of are two vectors, and . They span a parallelogram — and the difference ad-bc is exactly its signed area . The big ad rectangle, with the bc rectangle carved away, leaves precisely that parallelogram. The columns keep their counterclockwise order — orientation preserved. The columns are flipped — orientation reversed. The size |ad-bc| is still the area. The exact area of the image parallelogram — the factor by which A scales every area. The parallelogram collapses to a segment (area 0 ): the columns are linearly dependent, so A is singular — not invertible. You can now compute any determinant — diagonal product minus anti-diagonal product — and read what the number is telling you geometrically. Scroll up to revisit any step.

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