Home
Class 12
MATHS
If (i) A=[[cosalpha,sinalpha],[-sinalpha...

If (i) `A=[[cosalpha,sinalpha],[-sinalpha,cosalpha]]`, then verify that `AprimeA = I`.
(ii) `A=[[sinalpha,cosalpha],[-cosalpha,sinalpha]]`, then verify that `AprimeA = I`.

Text Solution

Verified by Experts

(i) Given `A=[[cosalpha,sinalpha],[-sinalpha,cosalpha]]`
`therefore`
`A'=[[cosalpha,-sinalpha],[sinalpha,cosalpha]]`
`A'A=[[cosalpha,-sinalpha],[sinalpha,cosalpha]] [[cosalpha,sinalpha],[-sinalpha,cosalpha]]`
`A'A=[[(cosalpha)(cosalpha)+(-sinalpha)(-sinalpha),(cosalpha)(sinalpha)+(-sinalpha)(cosalpha)],[(sinalpha)(cosalpha)+(cosalpha)(-sinalpha),(sinalpha)(sinalpha)+(cosalpha)(cosalpha)]] `
`A'A=[[cos^2alpha+sin^2alpha,sinalphacosalpha -sinalphacosalpha],[sinalphacosalpha-sinalphacosalpha,sin^2alpha+cos^2alpha ]] `
`A'A=[[1,0],[0,1]]=I`
Hence, we verified that `AprimeA = I`

(ii) `A=[[sinalpha,cosalpha],[-cosalpha,sinalpha]]`
`A'=[[sinalpha,-cosalpha],[cosalpha,sinalpha]]`
`A'A= [[sinalpha,-cosalpha],[cosalpha,sinalpha]][[sinalpha,cosalpha],[-cosalpha,sinalpha]]`

`A'A=[[(sinalpha)(sinalpha)+(-cosalpha)(-cosalpha),(sinalpha)(cosalpha)-(-cosalpha)(sinalpha)],[(sinalpha)(cosalpha)+(sinalpha)(-cosalpha),(cosalpha)(cosalpha)+(sinalpha)(sinalpha)]]`
`A'A=[[sin^2alpha+cos^2alpha,sinalphacosalpha -sinalphacosalpha],[sinalphacosalpha-sinalphacosalpha,cos^2alpha+sin^2alpha]] `

`A'A=[[1,0],[0,1]]=I`
Hence, we verified that `AprimeA = I`
Promotional Banner

Topper's Solved these Questions

Similar Questions

Explore conceptually related problems

If A=[[cosalpha, sinalpha], [-sinalpha, cosalpha]] , then A^(10)=

If A_(alpha)=[(cosalpha,-sinalpha),(sinalpha,cosalpha)] , then

If A=[(sinalpha,cosalpha),(-cosalpha,sinalpha)] , the prove that A'A=I .

if A=[[cosalpha,sinalpha],[-sinalpha,cosalpha]] be such that A+A'=I then alpha

If A=[{:(cosalpha,sinalpha),(-sinalpha,cosalpha):}] , show that A'A=I.

A = [ [ cosalpha , sinalpha ], [ sinalpha , cosalpha ] ] ,then find | A |

If A_(alpha)=[(cosalpha,sinalpha),(-sinalpha,cosalpha)] then (A_(alpha))^2=?