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

unnati dwivedi , 14 Years ago

Grade 12

2 Answers

Anil Pannikar AskiitiansExpert-IITB

Last Activity: 14 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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All the best.

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Anil Pannikar

IIT Bombay

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SAGAR SINGH - IIT DELHI

Last Activity: 14 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 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 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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Sagar Singh

B.Tech, IIT Delhi

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