16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.2.10

Determinants

Mathematics — Linear Algebra · rank-solution-space

Problem

Let FF be a field, AA a 2×32 \times 3 matrix over FF, and (c1,c2,c3)(c_1, c_2, c_3) the vector in F3F^3 defined by c1=∣A12A13A22A23∣,c2=∣A13A11A23A21∣,c3=∣A11A12A21A22∣.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)) rank(A)=2\mathrm{rank}(A) = 2 if and only if (c1,c2,c3)≠0(c_1, c_2, c_3) \neq 0; ((b)) if AA has rank 22, then (c1,c2,c3)(c_1, c_2, c_3) is a basis for the solution space of the system of equations AX=0AX = 0.

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