Positive determinant sign in 3D
The projected arrows appear in the cyclic column order, so det(A) > 0. Click the image to open the full-size version.
Determinant Sign Visualization
The determinant of a 3 × 3 matrix is the signed volume of the parallelepiped formed by its columns. These visuals show how the sign can be read from orientation, projected cyclic order and cofactor expansion.
Negative determinant sign in 3D
The visible projected order is reversed relative to the cyclic column order, so det(A) < 0. Click the image to open the full-size version.
Cofactor expansion visualization
This diagram interprets cofactor expansion as a sum of signed projected-volume terms. Click the image to open the full-size version.
How to read the sign visually
Choose one column vector as the viewing vector. Project the two remaining column vectors onto the plane orthogonal to it, then look from the tip of the viewing vector toward the origin.
- Method 1: project a₁ and a₂ onto the plane perpendicular to a₃
- Method 2: project a₃ and a₁ onto the plane perpendicular to a₂
- Method 3: project a₂ and a₃ onto the plane perpendicular to a₁
If the projected arrows appear in the cyclic column order a₁, a₂, a₃, a₁, … then det(A) > 0. If the visible order is reversed, then det(A) < 0. If the projected arrows are linearly dependent, then det(A) = 0.
Why the projection methods agree
The three methods agree because cyclic column shifts preserve determinant sign in three dimensions. Changing from [a₁ | a₂ | a₃] to [a₃ | a₁ | a₂] or [a₂ | a₃ | a₁] takes two column exchanges, so the determinant sign is unchanged.
After such a cyclic relabeling, applying the first projection method becomes identical to applying the second or third method to the original matrix.
Connection with cofactor expansion
Cofactor expansion splits the determinant into signed terms. In the visual interpretation, each term combines one projected column component with the signed area of a projected minor.
For more details, see the Determinant sign chapter PDF.
Related visual: Determinant from elementary matrices
Related chapter: Determinant sign
Related work: GraphMath Linear Algebra
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