Q.Find the rank of the given matrix $\begin{bmatrix}3 & 1 & -5 & -1\\1 & -2 & 1 & -5\\1 & 5 & -7 & 2\end{bmatrix}$

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 …