Q.Find the rank of $\begin{pmatrix}2&4&6\\1&2&3\\3&6&9\end{pmatrix}$.

Rank of a Matrix

The rank of a matrix is the number of non-zero rows in its row-echelon form,

equivalently the order of its largest non-vanishing minor, equivalently the maximum

number of linearly independent rows (or columns). Elementary row/column operations do

not change the rank, so a matrix is reduced by such operations and the surviving

non-zero rows are counted.

Consequences used in problems: a matrix has rank $1$ exactly when every $2\times2$

minor vanishes — i.e. all rows are proportional; the given non-zero condition on

entries then forces relations among the unknowns. An $n\times n$ matrix has full rank

$n$ iff its determinant is non-zero. When a matrix has repeated or proportional rows

(for example all rows equal), its rank drops accordingly (a $3\times3$ all-ones matrix

has rank $1$). Reading the echelon form after the stated operations gives the rank

directly.

Matrix rank builds on the NCERT/CBSE Class 12 Mathematics "Matrices" and "Determinants" chapters and is an important topic for JEE Main, JEE Advanced and state CETs, extending the row-reduction techniques introduced there. "Rank of a matrix examples" and "matrices class 12 maths important questions" are common searches this concept addresses.