wiz-icon
MyQuestionIcon
MyQuestionIcon
2
You visited us 2 times! Enjoying our articles? Unlock Full Access!
Question

If A=⎢ ⎢0tanα2tanα20⎥ ⎥ and I is a 2×2 unit matrix , then prove that I+A=(IA)[cosαsinαsinαsinα]

Open in App
Solution

Since I=[1001] and given

A=⎢ ⎢0tanα2tanα20⎥ ⎥

I+A=[1001]+⎢ ⎢0tanα2tanα20⎥ ⎥

=⎢ ⎢1tanα2tanα21⎥ ⎥

R.H.S=(IA)[cosαsinαsinαsinα]

=⎢ ⎢0tanα2tanα20⎥ ⎥[cosαsinαsinαsinα]

=⎢ ⎢0tanα2tanα20⎥ ⎥⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢1tan2α21+tan2α22tanα21+tan2α22tanα21+tan2α21tan2α21+tan2α2⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Let tan(α2)=λ, then
R.H.S=[1λλ1]⎢ ⎢ ⎢ ⎢1λ21+λ22λ1+λ22λ1+λ21λ21+λ2⎥ ⎥ ⎥ ⎥

=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢1λ2+2λ21+λ22λ+λ(1λ2)1+λ2λ(1λ2)+2λ1+λ22λ2+1λ21+λ2⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢1+λ21+λ2λ(1+λ2)1+λ2λ(1+λ2)1+λ22λ2+1λ21+λ2⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=⎢ ⎢1tanα2tanα21⎥ ⎥ since λ=tanα2

=I+A

=LHS

I+A=(IA)[cosαsinαsinαsinα]

Hence Proved

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Algebraic Operations
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon