Rank, trace and determinant: what each one measures

Three scalars summarise a matrix, and each answers a different question: rank says how many independent directions survive, determinant says how volume scales, and trace says what the eigenvalues add to. Knowing which one to reach for settles most interview questions without any computation.

Quantity	Geometric meaning	Key identity
Rank	Dimension of the image — independent directions	$\mathrm{rank}(AB) \le \min(\mathrm{rank}\,A, \mathrm{rank}\,B)$
Determinant	Signed volume scaling of the unit cube	$\det(AB) = \det A \det B$
Trace	Sum of the diagonal, and of the eigenvalues	$\mathrm{tr}(AB) = \mathrm{tr}(BA)$

Table 1.1 · The three summaries. The trace cyclicity identity is the one that does real work: it lets you move factors around a product inside a trace, which is how most matrix-calculus derivations get unstuck.

The determinant as a volume. A matrix maps the unit cube to a parallelepiped, and the determinant is that shape’s signed volume. Everything follows: a determinant of zero means the cube was flattened into a lower dimension, so information was lost and the map cannot be inverted. A negative determinant means the orientation was flipped. And $\det(AB) = \det A \det B$ is just the statement that applying two maps scales volume twice. Reading it as a volume makes the algebraic facts obvious rather than memorised.

Proposition 1.2

Rank and the fundamental relation

Rank is the number of linearly independent rows, which is also the number of independent columns — a non-obvious fact that does a great deal of work. Rank–nullity then says $\mathrm{rank}(A) + \dim(\ker A) = n$ : every dimension of the input either survives into the image or is collapsed into the null space.

Holds when

A product cannot have higher rank than either factor, which is why $X^\top X$ is singular when $X$ has more columns than rows.
A sample covariance matrix from $T$ observations has rank at most $T-1$ after demeaning.
Adding a matrix can raise rank; multiplying never can.

Equation 1.3

The identities to have ready

Transpose and inverse both reverse the order of a product. Forgetting that is the single most common slip in a matrix derivation.

(AB)^\top = B^\top A^\top, \qquad (AB)^{-1} = B^{-1}A^{-1}, \qquad \det(A^{-1}) = \frac{1}{\det A}

$\mathrm{tr}(A) = \sum\lambda_i$: Trace is the sum of the eigenvalues.
$\det(A) = \prod\lambda_i$: Determinant is their product — so a zero eigenvalue means a zero determinant.

Figure 1.4 · Trace, determinant and rank are the same three numbers. Trace is their sum,

7

; determinant is their product,

0

; rank is how many are non-zero,

2

. One zero eigenvalue is a singular matrix, a non-trivial null space and a regression you cannot solve — three statements about the same bar.

Rank, trace and determinant: what each one measures

After this lesson you should be able to

Rank and the fundamental relation

The identities to have ready

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