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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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(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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