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Minors and Matrix Rank

  • The study of consistency in linear equation systems relies fundamentally on the structural properties of matrices, specifically their dimensions, minors, and ranks.

  • Consider a rectangular matrix AA of dimension m×nm \times n:

Am×n=(a11a12…a1na21a22…a2n⋮⋮⋱⋮am1am2…amn)A_{m \times n} = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix}

  • By selecting rr rows and rr columns from matrix AA where 0≤r≤min⁡{m,n}0 \le r \le \min\{m, n\}, a square submatrix A′A' of size r×rr \times r is formed by the elements situated at their intersections:

Ar×r′=(a11′a12′…a1r′a21′a22′…a2r′⋮⋮⋱⋮ar1′ar2′…arr′)A'_{r \times r} = \begin{pmatrix} a'_{11} & a'_{12} & \dots & a'_{1r} \\ a'_{21} & a'_{22} & \dots & a'_{2r} \\ \vdots & \vdots & \ddots & \vdots \\ a'_{r1} & a'_{r2} & \dots & a'_{rr} \end{pmatrix}

  • Definition of Minor of Order rr: The determinant of the square submatrix Ar×r′A'_{r \times r} extracted from Am×nA_{m \times n} is called a minor of order rr (or an rr-th order minor) of matrix AA.

  • The total number of possible rr-th order minors that can be constructed from an m×nm \times n rectangular matrix is given by the product of combinations:

Nminors=Cmr⋅CnrN_{\text{minors}} = C_m^r \cdot C_n^r

  • A minor of order 11 corresponds directly to an individual element of the matrix. Consequently, the