12 Determinant Properties That Appear on Every Exam: det(kA), det(A⁻¹), det(adj A) and the Traps

Why the properties hold: the determinant is an area factor

A 2×2 matrix turns the unit square into a parallelogram, and $|\det A|$ is its area (for 3×3, a volume). Seen this way most properties are obvious: stretch every side by 2 and the area grows by $2^2$, the volume by $2^3$.

Properties 1–6: row operations and special matrices

1. Transpose: $\det A^T = \det A$ — so every row rule also works for columns
2. Swap two rows: the sign changes
3. Multiply one row by $k$: the determinant is multiplied by $k$ (factors come out one row at a time)
4. Add a multiple of another row $R_i \leftarrow R_i + kR_j$: no change — the main tool of row reduction
5. A zero row, two equal rows or proportional rows: $\det = 0$
6. Triangular or diagonal matrix: $\det$ = product of the diagonal entries

Properties 7–10: constants, products, inverses and powers

7. $\det(kA) = k^n\det A$ — because $k$ sits in all $n$ rows
8. $\det(AB) = \det A\det B = \det(BA)$, even though $AB \ne BA$
9. $\det(A^{-1}) = \dfrac{1}{\det A}$ — and $A$ is invertible exactly when $\det A \ne 0$
10. $\det(A^m) = (\det A)^m$

Property 11: the adjugate, where students slip most

Taking determinants of $A\operatorname{adj}A = (\det A)I$ gives $\det A\cdot\det(\operatorname{adj}A) = (\det A)^n$:

Property 12: what has no formula

A bonus trick: if every row has the same sum, add all columns into the first column and take that sum out as a factor. For example, the matrix with $a$ on the diagonal and $b$ elsewhere has $\det = (a - b)^{n-1}(a + (n - 1)b)$.