p is some matrix of order n and a is another matrix of the same order then what is

1. pap'(p'=p transpose)

2.pap-1(p-1=p inverse)

Vinay S R, 13 years ago

Grade:

1 Answers

SAGAR SINGH - IIT DELHI

878 Points

13 years ago

Dear vinay,

The transpose of a matrix ${\bf A}$ , denoted by ${\bf A}^T$ , is obtained by switching the positions of elements $a_{ij}$ and $a_{ji}$ for all $i,j \in \{1,\cdots,n\}$ . In other words, the ith column of ${\bf A}$ becomes the ith row of ${\bf A}^T$ , or equivalently, the ith row of ${\bf A}$ becomes the ith column of ${\bf A}^T$ :

$\begin{displaymath}{\bf A}^T=[{\bf a}_1 \cdots {\bf a}_n]^T= \left[ \begin{array}{c} {\bf a}_1^T .. .. {\bf a}_n^T \end{array} \right] \end{displaymath}$

where vector ${\bf a}_i$ is the ith column of ${\bf A}$ and its transpose ${\bf a}_i^T$ is the ith row of ${\bf A}^T$ .

For any two matrices ${\bf A}$ and ${\bf B}$ , we have

$\begin{displaymath}({\bf A}{\bf B})^T={\bf B}^T {\bf A}^T \end{displaymath}$

If ${\bf AB}={\bf BA}={\bf I}$ , where ${\bf I}$ is an identity matrix:

$\begin{displaymath}{\bf I}=diag[1,\cdots,1]=\left[ \begin{array}{cccc} 1 & 0 &... ...\cdot & \cdot & \cdot 0 & 0 & \cdot & 1 \end{array} \right] \end{displaymath}$

then ${\bf B}={\bf A}^{-1}$ is the inverse of ${\bf A}$ . ${\bf A}^{-1}$ exists iff $det({\bf A}) \neq 0$ , i.e., $rank({\bf A}) = n$ .

For any two matrices ${\bf A}$ and ${\bf B}$ , we have

$\begin{displaymath}({\bf AB})^{-1}={\bf B}^{-1}{\bf A}^{-1} \end{displaymath}$

and

$\begin{displaymath}({\bf A}^{-1})^T=({\bf A}^T)^{-1} \end{displaymath}$

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.

Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.

Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar respectively : Click here to download the toolbar..

Askiitians Expert

Sagar Singh

B.Tech, IIT Delhi