Rank is defined for any matrix (need not be square).

**Some important concepts: **

**⇒ Sub matrix of a matrix:** Suppose A is any matrix of the type m x 72. Then a matrix obtained by leaving some rows and some columns from A is called sub-matrix of A.

**⇒ Rank of a Matrix:** A number r is said to be the rank of a matrix A, if it possesses the following properties:

⇒ There is at least one square sub-matrix of A of order r whose determinant is not equal to zero. b. If the matrix A contains any square sub-matrix of order (r + 1) and above, then the determinant of such a matrix should be zero.

Put together property (a) and (b) give the definition of the rank of a matrix as the “size of the largest non-zero minor”.

**Note: **

i. The rank of a matrix is ≤ r, if all (r + 1) – rowed minors of the matrix vanish.

ii. The rank of a matrix is ≤ r, if there is at least one r – rowed minor of the matrix which is not equal to zero.

iii. The rank of transpose of a matrix is same as that or original matrix.

i.e. r (A^{T}) = r (A).

iv. Rank of a matrix is same as the number of linearly independent row vectors in the matrix as well as the number of linearly independent column vectors in the matrix.

v. For any matrix A, rank (A) ≤ min (m, n)

i.e., minimum rank of A_{mxn }= min (m, n)

vi. Rank (AB) ≤ rank A

rank (AB) ≤ rank B

so, rank (AB) ≤ min(rank A, rank B)

vii. Rank (A^{t}) = Rank (A)

viii. Rank of a matrix is the number of non-zero rows in its echelon form.

ix. Elementary transformations do not alter the rank of a matrix.

x. Only null matrix can have a rank of zero. All other matrices have rank of at least one.

xi. Similar matrices have the same rank.

**Echelon form: **A matrix is in echelon form if only if

⇒ Leading non-zero element in every row is behind leading non-zero element in previous row. This means below the leading non-zero element in every row all the element must be zero.

⇒ All the zero rows should be below all the non-zero rows. This definition gives an alternate way of calculating the rank of larger matrices (larger 3 ? 3) more easily. To reduce a matrix t its echelon form use gauss elimination method on the matrix and convert it into an upper triangular matrix. Which will be echelon form. Then count the number of non-zero rows in the upper triangular matrix to get the rank of the matrix.

**Example:** Find the rank of matrix \(\left[ A \right]=\left[ \begin{matrix}4 & 2 & 1 & 3 \\6 & 3 & 4 & 7 \\2 & 1 & 0 & 1 \\\end{matrix} \right]\).

**Solution: **Consider first 3 x 3 minors, since maximum possible rank is 3.

\(\left| \begin{matrix}4 & 2 & 1 \\6 & 3 & 4 \\2 & 1 & 0 \\\end{matrix} \right|=0\)

\(\left| \begin{matrix}2 & 1 & 3 \\3 & 4 & 7 \\1 & 0 & 1 \\\end{matrix} \right|=0\)

and \(\left| \begin{matrix}4 & 2 & 3 \\6 & 3 & 7 \\2 & 1 & 1 \\\end{matrix} \right|=0\)

Since all 3 x 3 minors are zero, now try 2 x 2 minors.

\(\left| \begin{matrix}4 & 2 \\6 & 3 \\\end{matrix} \right|=0M\)

So, \(\left| \begin{matrix}2 & 1 \\3 & 4 \\\end{matrix} \right|\) = 8 – 3 = 5 ≠ 0.

Rank = 2.