Kunze Linear Algebra · Problem 5.2.10

Determinants

Mathematics — Methods · rank-determinants

Problem

Let

F

be a field,

A

2 \times 3

matrix over

F

, and

(c_1, c_2, c_3)

the vector in

F^3

defined by

c_1 = \begin{vmatrix}A_{12} & A_{13}\\ A_{22} & A_{23}\end{vmatrix}, \quad c_2 = \begin{vmatrix}A_{13} & A_{11}\\ A_{23} & A_{21}\end{vmatrix}, \quad c_3 = \begin{vmatrix}A_{11} & A_{12}\\ A_{21} & A_{22}\end{vmatrix}.

Show that ((a))

\text{rank}(A) = 2

if and only if

(c_1, c_2, c_3) \neq 0

; ((b)) if

A

has rank

2

, then

(c_1, c_2, c_3)

is a basis for the solution space of the system of equations

AX = 0

Intuitive explanation

The vector $(c_1, c_2, c_3)$ is built from all possible $2 \times 2$ minors of $A$ —these capture the "obstruction" to $A$ having full rank. The key insight is that $A$ has rank 2 exactly when at least one minor is nonzero, and when that happens, this vector encodes the left nullspace of $A$ : it's orthogonal to every row of $A$ , making it a natural candidate for solving $AX = 0$ . Verifying the orthogonality and dimension will establish both parts.

Intuitive explanation

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