Rank of a Matrix
An important characteristic of any matrix is its rank. It tells us the number of independent rows or columns of matrix.
Note that all matrices have a rank, unlike inverse of a matrix, which only a non-singular square matrix has.
Definition: A matrix has a rank , if
I) At least one minor of order which is not equal to and
II) Every minor of order is .
[A minor of a matrix is the determinant of some smaller square matrix, formed by removing one or more rows or columns of . ]
The elementary transformations of a matrix do not alter its rank. Any matrix , obtained by transforming a matrix is known as equivalent matrix. It is denoted by .
Rank of a Matrix by reducing it to Echelon/ Canonical Form
A matrix is in echelon/ canonical form if
I) all nonzero rows are above any rows of all zeroes (if any), and
II) the leading coefficient (the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it. The pivot is preferably taken to be .
The rank is the number of non-zero rows of matrix.
Rank of a Matrix by reducing it to Normal Form
A normal form of a non-zero matrix is a matrix in either of the following forms:
is identity matrix of order .
We use both and transformations to reduce the matrix to normal form. The rank of the matrix is then equal to .
Obtaining 2 matrices and , such that is in normal form
Consider a matrix with rows and columns. Take as and as . We can then write
Reduce on the LHS to a normal form. Perform respective row transformations on and column transformations on . Once is reduced to normal form, we get the rank.
Note: and are not unique, and are non-singular.
Inverse using form
If is in normal form, is .