# Rank of a Matrix

• ### 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 ${r}$, if

I) At least one  minor of order ${r}$ which is not equal to ${0}$ and

II) Every minor of order ${r+1}$ is ${0}$.

[A minor of a matrix ${A}$ is the determinant of some smaller square matrix, formed by removing one or more rows or columns of ${A}$. ]

The elementary transformations of a matrix do not alter its rank. Any matrix ${B}$, obtained by transforming a matrix ${A}$ is known as equivalent matrix. It is denoted by ${A \sim B}$.

• ### 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 ${1}$.

The rank ${r}$ 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 ${A}$ is a matrix in either of the following forms:

${[I_r], \ \begin {bmatrix} I_r & 0 \end {bmatrix}, \ \begin {bmatrix} I_r \\ 0 \end{bmatrix}, \ \begin {bmatrix} I_r & 0 \\ 0 & 0 \end {bmatrix}}$

${I_r}$ is identity matrix of order ${r}$.

We use both ${row}$ and ${column}$ transformations to reduce the matrix to normal form. The rank of the matrix is then equal to ${r}$.

• ### Obtaining 2 matrices ${P}$ and ${Q}$, such that ${PAQ}$ is in normal form

Consider a matrix ${A}$ with ${m}$ rows and ${n}$ columns. Take ${P}$ as ${I_m}$ and ${Q}$ as ${I_n}$. We can then write

${A_{m \times n} = I_{m} \times A \times I_n = PA Q}$

Reduce ${A}$ on the LHS to a normal form. Perform respective row transformations on ${I_m}$ and column transformations on ${I_n}$. Once ${A}$ is reduced to normal form, we get the rank.

Note: ${P}$ and ${Q}$ are not unique, and are non-singular.

• ### Inverse using ${PAQ}$ form

If ${PAQ}$ is in normal form, ${A^{-1}}$ is ${QP}$.