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Rank of a square matrix is found by finding the determinant. If the determinant is non-zero, the dimension of the square matrix is the rank.

If the determinant of A of size n x n is zero, then it is possible that there are two identical rows or columns. Or, algebraic sum of some rows or columns is equal to a row or column. Then the rank cannot be n.

We remove one of the identical rows or columns and then find the determinant of n-1 x n-1 matrix. If its determinant is non zero, then n-1 will be the rank of the matrix A.

Determinant is found by the usual method.

det|  \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] | = det | \left[\begin{array}{ccc}1&2&3\\4-1&5-2&6-3\\7-1&8-2&9-3\end{array}\right] | \\ \\ \\ = det| \left[\begin{array}{ccc}1&2&3\\3&3&3\\6&6&6\end{array}\right] | = 0 \\ \\
Determinant is zero because 3nd row is a multiple of 2nd row.

Let us remove the third row and column. and find the determinant again of the 2x2 matrix.

det|  \left[\begin{array}{ccc}1&2\\4&5\end{array}\right] | = 1*5 - 4*2 = -3  \neq 0. \\ \\ So\ Rank = 2.\\

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