13.20 RANK OF A MATRIX
Definition 13.86. A matrix is said to be of rank r if it has at least one non-singular submatrix of order r but has no non-singular submatrix of order more than r.
Rank of a matrix A is denoted by ρ(A).
A matrix is said to be of rank zero if and only if all its elements are zero.
Find the rank of the matrix
Solution. The matrix A is of order 3×4. Therefore, ρ(A) ≤ 3. We note that
Therefore, ρ(A) ≠ 3. But, we have submatrix , whose determinant is equal to –2 ≠ 0. Hence, by definition, ρ(A) = 2.