3.2. The rank of a matrix: computation

1) in each row, the first nonzero entry is 1 and all the other entries in that column are zero.

2) a row in which all the entries are zero, is below all those rows which have nonzero entries.

3) if the rows r₁ and r₂ , r₁ < r₂ , have their first nonzero elements in the columns c₁ and c₂ , respectively, then c₁ < c₂ .

Example 1: A matrix in reduced echelon form

Proposition 4. Every matrix A is equivalent to a unique matrix A_R , which is in reduced echelon form. The rank of A = the rank of A_R = the number of nonzero rows in A_R .

Proof. Omitted.

Proposition 5. If A is an n x n matrix, then

Proof.
1) A_R = I   =>    the rank of A = n (= the number of nonzero rows)
2) The rank of A = n   =>    the rank of A_R = n   =>    A_R = I.

Example 2: Rank, nullity, kernel, basis for a nullspace