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Home » Math Homework Help » Algebra Homework Help » Rank of a Matrix
Rank of a Matrix
Definition: In a matrix A of order (m × n) if,

(i) At least one minor of order r is not zero,

(ii) Every minor of order (r + 1) is zero,

Then r is said to be the rank of the given matrix A and is denoted by ρ(A).

Since every minor of order (r + 2) of A can be expressed as a linear combination of the minors of order (r + 1) of A, therefore, we can say that if ρ(A) = r, then every minor of order (r + 2) will also be zero. Similarly, it can be seen that every minor of order (r + 3), (r + 4), … etc will also be zero. Thus we observe that if ρ(A) = r then

(i) At least one minor of order r is not zero,

(ii) Every minor of order (r + 1) is zero,

In other words, the rank of a matrix is the largest order of any non-vanishing minor.

Note 1: If every minor of order greater than or equal to r is zero, than ρ(A) < r.

Note 2: If there exists a non-zero minor of order r, then ρ(A) ≥ r.

Note 3: For a non-zero matrix, the least value of its rank is one.

Note 4: We agree to define rank of a zero matrix as zero.

Note 5:
Rank of every non-singular matrix of order n is n.

Note 6: ρ(In) = n, where In is a unit of matrix of order n.

Note 7: Obviously, ρ(A’) = ρ(A) and ρ(A0) = ρ(A).

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