The determinant is a single number calculated from a square matrix, and it tells you how that matrix changes space. For a 2 by 2 matrix, the determinant measures the signed area scale factor of the transformation. For a 3 by 3 matrix, it measures the signed volume scale factor.
This makes determinants useful in geometry, linear algebra, physics, computer graphics, and systems of equations.
A determinant also tells whether a matrix is invertible. If det(A) = 0, the transformation squashes space into a lower dimension, so information is lost and the matrix has no inverse. If det(A) is nonzero, the transformation can be undone.
The sign of the determinant shows whether orientation is preserved or flipped.
Key Facts
- For A = [[a, b], [c, d]], det(A) = ad - bc.
- For a 2 by 2 matrix, |det(A)| is the area scale factor of the linear transformation.
- For a 3 by 3 matrix, |det(A)| is the volume scale factor of the linear transformation.
- A matrix A is invertible if and only if det(A) != 0.
- For 3 by 3 matrices, det(A) can be found by cofactor expansion: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg).
- The product rule says det(AB) = det(A)det(B).
Vocabulary
- Determinant
- A number assigned to a square matrix that describes area or volume scaling and whether the matrix is invertible.
- Square matrix
- A matrix with the same number of rows and columns.
- Invertible matrix
- A square matrix that has an inverse matrix, meaning its transformation can be undone.
- Area scale factor
- The number by which a transformation multiplies the area of a shape.
- Orientation
- The handedness or direction order of a shape, which can be preserved or flipped by a transformation.
Common Mistakes to Avoid
- Using ad + bc for a 2 by 2 determinant is wrong because the correct formula is ad - bc.
- Ignoring the sign of the determinant is wrong because a negative determinant means the transformation flips orientation.
- Thinking det(A) = 0 means the matrix has no solutions is wrong because it means the matrix is not invertible, while a system may have no solutions or infinitely many solutions depending on the constants.
- Applying determinant rules to non-square matrices is wrong because determinants are defined only for square matrices.
Practice Questions
- 1 Compute the determinant of A = [[3, 5], [2, 7]]. Then state the area scale factor.
- 2 Compute det(B) for B = [[1, 2, 0], [3, -1, 4], [2, 1, 5]].
- 3 A 2 by 2 matrix has determinant -4. Explain what this means for area, orientation, and invertibility.