Traps in Matrices & Determinants

For an $n \times n$ matrix $A$, $|kA| = k^n|A|$, not $k|A|$.

Why: The scalar multiplies each of the $n$ rows, so it appears $n$ times in the determinant.

$|A + B|
eq |A| + |B|$ in general. This is a very common error.

Why: Determinant is multiplicative ($|AB| = |A||B|$) but NOT additive.

$D = 0$ does not automatically mean infinite solutions. You need all $D_i = 0$ as well.

Why: Students memorize 'D = 0 means infinite' without checking the numerator determinants.

$\text{adj}(\text{adj}(A)) = |A|^{n-2} \cdot A$, and $|\text{adj}(A)| = |A|^{n-1}$, not $|A|^n$.

Why: The exponent $(n-1)$ is easy to confuse with $n$ or $(n-2)$ in different formulas.

$AB
eq BA$ in general. Only commutative in special cases.

Why: Real number intuition ($ab = ba$) does not extend to matrices.