Determinants
The idea
The determinant compresses a square matrix into one number with a geometric meaning: the factor by which the matrix scales area (in 2 × 2) or volume (in 3 × 3), with a negative sign recording an orientation flip. That single number answers structural questions instantly — a matrix is invertible exactly when its determinant is nonzero, which is also exactly when the corresponding linear system has a unique solution and the columns are linearly independent.
For computation, cofactor expansion works along any row or column: each entry multiplies the determinant of the smaller matrix left after deleting its row and column, with signs alternating in a checkerboard pattern. Strategy matters — expand along the row or column with the most zeros, since each zero kills a whole term. For larger matrices, row reduction is faster, tracking how each row operation affects the determinant.
The misconception that costs the most points: the determinant is not linear over addition, so det(A + B) is generally not det(A) + det(B). The reliable rules are multiplicative ones — det(AB) = det(A)det(B), and scaling a single row scales the determinant by that factor, so scaling an entire n × n matrix by c scales the determinant by c to the power n.
Worked example
Compute the determinant of the 3 × 3 matrix A with rows (2, 1, 3), (0, 4, −1), and (5, 2, 1), and state whether A is invertible.
- Choose the expansion line strategically: column 1 contains a zero (the entry 0 in row 2), so expanding down column 1 leaves only two terms to compute.
- First term: entry 2 at position (1,1) carries sign +, times the minor from deleting row 1 and column 1, det of rows (4, −1) and (2, 1), which is 4(1) − (−1)(2) = 6. Contribution: 2 × 6 = 12.
- The entry 0 at position (2,1) contributes nothing — exactly why the column was chosen.
- Third term: entry 5 at position (3,1) carries sign + (checkerboard: row 3, column 1), times det of rows (1, 3) and (4, −1), which is 1(−1) − 3(4) = −13. Contribution: 5 × (−13) = −65.
- Total: det(A) = 12 + 0 − 65 = −53. Double-check by expanding along row 1: 2(4 + 2) − 1(0 + 5) + 3(0 − 20) = 12 − 5 − 60 = −53 — same value, as expansion along any line must give.
- Interpret: det(A) = −53 ≠ 0, so A is invertible, its columns are independent, and as a map it scales volume by 53 while reversing orientation.
Answer. det(A) = −53, so A is invertible and scales volumes by a factor of 53 with an orientation flip.
Check your understanding
- Why does a zero determinant mean the matrix squashes space into a lower dimension?
- How does the option to expand along any row or column help you pick the least painful computation?
- Why does det(AB) = det(A)det(B) make sense if you think of determinants as volume-scaling factors?
- What happens to a determinant under each of the three row operations, and how does that turn row reduction into a determinant algorithm?
Build the foundations first
Determinants builds on these concepts. If any feel shaky, start there.