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.

EXAMPLE 13.43

Find the rank of the matrix

 

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Solution.   The matrix A is of order 3×4. Therefore, ρ(A) ≤ 3. We note that

 

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Therefore, ρ(A) ≠ 3. But, we have submatrix , whose determinant is equal to –2 ≠ 0. Hence, by definition, ρ(A) = 2.

EXAMPLE ...

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