Given $A = \left[ {\begin{array}{*{20}{c}}2&3&1&4\\0&1&2&{ - 1}\\0&{ - 2}&{ - 4}&2\end{array}} \right],$
$({R_2} \to 2{R_2} + {R_3})$
$A = \left[ {\begin{array}{*{20}{c}}2&3&1&4\\0&0&0&0\\0&{ - 2}&{ - 4}&2\end{array}} \right]$
Since every minor of order $3$ in $ A$ is $0$ and there exists a minor order $3 $
i.e. $\left[ {\begin{array}{*{20}{c}}2&3\\0&{ - 2}\end{array}} \right]$ in $ A$ which is non $-$ zero.
Thus, rank $= 2.$
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જો કોઈક વાસ્તવિક સંખ્યા $\alpha$ અને $\beta$ માટે આપલે સમતલો $x+4 y-2 z=1$ ; $x+7 y-5 z=\beta$ ; $x+5 y+\alpha z=5$ નો છેદગણ અવકાશમાં રેખા દર્શાવે છે તો $\alpha+\beta$ મેળવો.
જો $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in R$ અને $A^{4}=\left[a_{i j}\right]$ તથા $a_{11}=109,$ હોય તો $a_{22}$ ની કિમત શોધો