Algebra> rank of a matrix...

How to find the rank of a matrix??

Naina Razdan , 14 Years ago

Grade 12

1 Answers

SAGAR SINGH - IIT DELHI

Last Activity: 14 Years ago

Dear eshita,

The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns). A square matrix A_n×n is non-singular only if its rank is equal to n.

Determine the row-rank of

$A = \begin{bmatrix}1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix}.$
Solution: To determine the row-rank of we proceed as follows.

$\begin{bmatrix}1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix} \overrightarr... ..._{31}(-1)} \begin{bmatrix}1 & 2 & 1 \\ 0 & -1 & -1 \\ 0 & -1 & 1 \end{bmatrix}.$
$\begin{bmatrix}1 & 2 & 1 \\ 0 & -1 & -1 \\ 0 & -1 & 1 \end{bmatrix} \overright... ..., R_{32}(1) } \begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \end{bmatrix}.$
$\begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \end{bmatrix}\overrightarro... ...R_{12}(-2) } \begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}.$
$\begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\overrightarr... ...1), R_{13}(1)}\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

The last matrix is the row reduced form of which has non-zero rows. Thus,

${\mbox{row-rank}}(A)~=~3.$

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