##### 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

**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.

**EXAMPLE ...**

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