Rank of a Matrix:
Rank is defined for any matrix Amxn (need not be square)
Submatrix of a Matrix: Suppose A is any matrix of the type m ´ n. 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.
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.
Important points:
- The rank of a matrix is ≤ r, if all (r + 1) – rowed minors of the matrix vanish.
- 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.
- The rank of transpose of a matrix is same as that or original matrix. i.e. r (At) = r (A).
- 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.
- For any matrix A, rank (A) ≤ min (m,n)
- e., maximum rank of Amxn = min(m, n)
- Rank (AB) ≤rank B so, rank (AB) ≤ min(rank A, rank B)
- Rank (At) = Rank (A)
- Rank of a matrix is the number of non-zero rows in its echelon form.
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.
- Elementary transformations do not alter the rank of amatrix.
- Only null matrix can have a rank of zero. All other matrices have rank of atleast one.
- Similar matrices have the same rank.
Example: Find the rank of matrix [A] = \(\left[ \begin{matrix} 4 & 2 & 1 & 3 \\ 6 & 3 & 4 & 7 \\ 2 & 1 & 0 & 1 \\\end{matrix} \right]\).
Solution: Consider first 3 ´ 3 minors, since maximum possible rank is 3
\(\left| \begin{matrix} 4 & 2 & 1 \\ 6 & 3 & 4 \\ 2 & 1 & 0 \\\end{matrix} \right|=0\),
\(\left| \begin{matrix} 2 & 1 & 3 \\ 3 & 4 & 7 \\ 1 & 0 & 1 \\\end{matrix} \right|=0\),
\(\left| \begin{matrix} 4 & 1 & 3 \\ 6 & 4 & 7 \\ 2 & 0 & 1 \\\end{matrix} \right|=0\),
And \(\left| \begin{matrix} 4 & 2 & 3 \\ 6 & 3 & 7 \\ 2 & 1 & 1 \\\end{matrix} \right|=0\).
Since all 3 ´ 3 minors are zero, now try 2 ´ 2 minors.
\(\left| \begin{matrix} 4 & 2 \\ 6 & 3 \\\end{matrix} \right|=0\),
So, \(\left| \begin{matrix} 2 & 1 \\ 3 & 4 \\\end{matrix} \right|=8-3=5\ne 0\),
Rank = 2.