Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 10

Need solution for RD Sharma Maths Class 12 Chapter Algebra of Matrices Excercise 4.4 Question 10

Answers (1)

Answer: AA^{T}=I

Given: If l_{i},m_{i},n_{i},i=1,2,3 denotes direction cosines of 3 mutually ⊥ vertices

A=\begin{bmatrix} l_{1} &m_{1} &n_{1} \\ l_{2} & m_{2} & n_{2}\\ l_{3} & m_{3} & n_{3} \end{bmatrix}

Hint: I refers to identity function, theA^{T} of matrix A can be obtained by reflecting the elements along its main diagonal.

Solution:

Given\left ( l_{1}, m_{1}, n_{1} \right ),\left ( l_{2}, m_{2}, n_{2}\right ), (l_{3}, m_{3}, n_{3}) are the direction cosines of 3 mutually + vectors in space.

               \begin{Bmatrix} l_{1}^{2}+ m_{1}^{2}+ n_{1}^{2}=1 \\ l_{2}^{2}+ m_{2}^{2}+ n_{2}^{2}=1\\ l_{3}^{2}+m_{3}^{2}+ n_{3}^{2} =1 \end{Bmatrix}                                                                                                 ……. (1)

               \begin{Bmatrix} l_{1}l_{2}+ m_{1}m_{2}+ n_{1}n_{2}=0 \\ l_{2}l_{3}+ m_{2}m_{3}+ n_{2}n_{3}=0\\ l_{3}l_{1}+m_{3}m_{1}+ n_{3}n_{1} =0 \end{Bmatrix}                                                                                      ……….. (2)

Let, A=\begin{bmatrix} l_{1} &m_{1} &n_{1} \\ l_{2} & m_{2} & n_{2}\\ l_{3} & m_{3} & n_{3} \end{bmatrix},A^{T}=\begin{bmatrix} l_{1} &l_{2} &l_{3} \\ m_{1} & m_{2} & m_{3}\\ n_{1} & n_{2} & n_{3} \end{bmatrix}

               AA^{T}=\begin{bmatrix} l_{1} &m_{1} &n_{1} \\ l_{2} & m_{2} & n_{2}\\ l_{3} & m_{3} & n_{3} \end{bmatrix}\begin{bmatrix} l_{1} &l_{2} &l_{3} \\ m_{1} & m_{2} & m_{3}\\ n_{1} & n_{2} & n_{3} \end{bmatrix}

               AA^{T}=\begin{bmatrix} l_{1}^{2}+m_{1}^{2} +n_{1}^{2} & l_{1} l_{2}+m_{1}m_{2}+n_{1}n_{2} & l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}\\ l_{1} l_{2}+m_{1}m_{2}+n_{1}n_{2} & l_{2}^{2}+m_{2}^{2} +n_{2}^{2} &l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3} \\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1} &l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3} & l_{3}^{2}+m_{3}^{2}+n_{3}^{2} \end{bmatrix}

From (1) & (2) we get

               AA^{T}=\begin{bmatrix} 1 & 0 &0 \\ 0& 1 & 0\\ 0 &0 &1 \end{bmatrix}=I

Hence, proved.

Posted by

infoexpert27

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

PM Modi's Pariksha Pe Charcha 2025 to be held next month, registrations open till January 14
anam-khan

CBSE Board Exams 2025: Online portal for availing exemptions by CWSN students opens
anam-khan

Board Exams: States agree to equivalence; no question paper ‘jumbling’ from next year, says PARAKH CEO

Latest Question