LA2.2.5

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Rank of A\leq 3\, since A is of 3rd order.

|A|=4(-6+6)-2(-12+12)+3(-8+8)=0\,

Since |A|=0\, rank of A<3\, i.e r(A)\leq 2\,

Consider the determinants of 2nd order submatrices

{\begin{vmatrix}4&2\\8&4\end{vmatrix}}=0\,

{\begin{vmatrix}2&3\\4&6\end{vmatrix}}=0\,

{\begin{vmatrix}4&3\\8&6\end{vmatrix}}=0\,

{\begin{vmatrix}4&2\\-2&-1\end{vmatrix}}=0\,

{\begin{vmatrix}4&3\\-1&-15\end{vmatrix}}=0\,

{\begin{vmatrix}4&3\\-2&-15\end{vmatrix}}=0\,

Since all 2nd order submatrices have zero determinants i.e 2nd order minors are all zero.

So r(A)<2\,

Since A is a non-zero matrix r(A)>0\,

Thus the rank of A is 1.

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