Discussió rang d’una matriu

Sea la matriu $$A = \begin{pmatrix}1 & m & -1 & 3 \\ m & 1 & 2 & m \\ -6 & 3 & -14 & m\end{pmatrix}$$ Calcular el rang de $A$ per als diferents valors de $m$.

$$|A_1| =
\begin{vmatrix}
1 & m & -1 \\
m & 1 & 2 \\
-6 & 3 & -14
\end{vmatrix}
= 14m^2 – 15m – 26 = 0 \quad \Rightarrow \quad m = 2, \quad m = -\frac{13}{4}$$

$$|A_2| =
\begin{vmatrix}
1 & m & 3 \\
m & 1 & m \\
-6 & 3 & m
\end{vmatrix}
= -m^3 – 6m^2 + 7m + 18 = 0 \quad \Rightarrow \quad m = 2, \quad m = -4 \pm \sqrt{7}$$

$$|A_3| =
\begin{vmatrix}
1 & -1 & 3 \\
m & 2 & m \\
-6 & -14 & m
\end{vmatrix}
= m^2 – 20m + 36 = 0 \quad \Rightarrow \quad m = 2, \quad m = 18$$

$$|A_4| =
\begin{vmatrix}
m & -1 & 3 \\
1 & 2 & m \\
3 & -14 & m
\end{vmatrix}
= 16m^2 – 2m – 60 = 0 \quad \Rightarrow \quad m = 2, \quad m = -\frac{15}{8}$$

Si $m \neq 2 \quad \Rightarrow \quad \text{Rang}(A) = 3$.

Quan $m = 2 ) \quad \Rightarrow \quad \text{Rang}(A) = 2, \text{ ja que el menor determinant }$
$$\begin{vmatrix}
1 & 2 \\
3 & 14
\end{vmatrix}
= 8 \neq 0.$$