Determinants and inverses · Grade 12
In Grade 12 you learn: inverse matrices.
The determinant of a square matrix is a single number. It tells you how the matrix scales area (or volume) and whether it can be undone. If the determinant is not zero, the matrix has an inverse. The inverse solves matrix equations and reverses transformations.
Key facts
- det [[a, b], [c, d]] = ad − bc.
- |det| is the area scale factor. A negative determinant means the orientation is reversed (a reflection is involved).
- det = 0: the matrix is singular and has no inverse. It squashes the plane onto a line or a point.
- The inverse of [[a, b], [c, d]] is 1/(ad − bc) [[d, −b], [−c, a]].
- det(AB) = det A × det B, and (AB)^(−1) = B^(−1)A^(−1).
- 3 × 3: expand along a row with the sign pattern + − +: a(ei − fh) − b(di − fg) + c(dh − eg).
Worked examples
Example 1
Find the determinant and the inverse of [[4, 3], [2, 2]].
Answer: det = 2, inverse [[1, −1.5], [−1, 2]]