# Determining the Rank of a Matrix A=((1,2,4),(3,6,5))

Determining the Rank of a Matrix
$A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$
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The matrix has $2$ rows and $3$ columns, so the greatest possible value of its rank is $2$. We pick any element which is not $0$.
$\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$
We form an order $2$ minor containing $1$.
$\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$
We calculate this minor.
$\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right]=6-6=0$
We form another order $3$ minor containing $1$. $A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$
We calculate this minor.
$\left[\begin{array}{cc}1& 4\\ 3& 5\end{array}\right]=5-12=-7\ne 0$
The rank is $2$