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Naina Razdan Grade: 12
        

How to find the rank of a matrix??

7 years ago

Answers : (1)

SAGAR SINGH - IIT DELHI
879 Points
										

Dear eshita,


The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns). A square matrix An×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 $ A,$ we proceed as follows.



  1. $ \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}.$

  2. $ \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}.$

  3. $ \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}.$

  4. $ \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 $ A$ which has $ 3$ non-zero rows. Thus,



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


 



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7 years ago
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