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wat do trace and rank of a matrix mean?????

wat do trace and rank of a matrix mean?????

Grade:12

2 Answers

Anil Pannikar AskiitiansExpert-IITB
85 Points
13 years ago

Dear Unnati,

 Trace : Sum of the elements of A (square matrix of order n ) lying along the principle diagonal is called trace of A.

Ex -   matrix =  1    0     so trace = 1+5 =6

                       0   5

Rank : No. r is said to be the rank of a m*n matrix A if,

1 every square sub matrix of order (r+1) or more is singular and

2 there exists at least one square submatrix of order r which is non singular

or rank of a matrix is the order of highest order non singular sub matrix

Ex - A = 1    2    3

            4    5    6

            3    4    5

we have det. A =0 so, r(A) is less then 3, also we have   5    6    a non singular square sub matrix of order 2 .

                                                                               4    5

hence rank of A = 2

 

 

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Askiitians Expert

Anil Pannikar

IIT Bombay

 

 

SAGAR SINGH - IIT DELHI
878 Points
13 years ago

Dear unnati,

Let ${\bf A}$ be an $n \times n$ square matrix:

\begin{displaymath}{\bf A}=\left[ \begin{array}{cccc} a_{11} & a_{12} & \cdot &... ...n1} & a_{n2} & \cdot & a_{nn} \end{array} \right]_{n\times n} \end{displaymath}


where

\begin{displaymath}\left[ \begin{array}{c} a_{1j} a_{2j} ... a_{nj} \end{array}\right] \end{displaymath}


is the jth column vector and

\begin{displaymath}[a_{i1}\;a_{i2}\;\cdots\;a_{in}]\end{displaymath}


is the ith row vector.

The n rows span the row space of ${\bf A}$ and the n columns span the column space of ${\bf A}$. The dimensions of these two spaces are the same and called the rank of ${\bf A}$:

\begin{displaymath}R=rank({\bf A}) \leq N \end{displaymath}

 

The determinant of $A$ is denoted by

\begin{displaymath}det({\bf A}) = \vert {\bf A} \vert \end{displaymath}


and we have

\begin{displaymath}\vert {\bf A}{\bf B} \vert=\vert{\bf A}\vert\; \vert{\bf B}\vert \end{displaymath}

 

$rank({\bf A}) < N$ if and only if

$det({\bf A})=0$.

The trace of $A$ is defined as the sum of its diagonal elements:

\begin{displaymath}tr({\bf A}) = \sum_{i=1}^n a_{ii} \end{displaymath}

Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.

All the best.

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Askiitians Expert

Sagar Singh

B.Tech, IIT Delhi

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