Matrix Rank and Elementary Row Operations, Slides of Mathematics

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The rank of a matrix

Lê Xuân Trường

Minors of order k

Definition

Let A be an m × n - matrix. A minor of order k of A is a determinant of a

k × k sub-matrix of A.

We obtain the minors of order k from A by first deleting m − k rows and

n − k columns, and then computing the determinant. There are usually

many minors of A of a given order.

Example: Find the minors of order 3 of the matrix

Computing the rank

Start with the minors of maximal order k. If there is one that is non-zero,

then rank(A) = k. If all maximal minors are zero, then rank(A) < k, and

we continue with the minors of order k1 and so on, until we find a minor

that is non-zero. If all minors of order 1 (i. e. all entries in A) are zero the

rank(A) = 0.

Example: Find the rank of the matrix

Properties

rank(A) ≤ min{m, n}

rank(A

T ) = rank(A)

If A is a square matrix of order n then

A is invertible ⇐⇒ det(A) 6 = 0 ⇐⇒ rank(A) = n

The elementary row operation do not change the rank of a matrix.

Find the rank of a matrix using elementary row operations

A

elementary operations −−−−−−−−−−−→ B (row echelon form)

=⇒ rank(A) = number of non-zero rows of B

Example: Find the rank of matrix A =

A −→

=⇒ rank(A) = 3