If `A(theta)=[(sin theta, i cos theta),(i cos theta, sin theta)]`, then which of the following is not true ?

Monjur Alahi

If `A(theta)=[(sin theta, i cos theta),(i cos theta, sin theta)]`, then which of the following is not true ?
A. `A(theta)^(-t)=A(pi-theta)`
B. `A(theta)+A(pi+theta)` is a null matrix
C. `A(theta)` is invertible for all `theta in R`
D. `A(theta)^(-1)=A(-theta)`


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Salim Ahmed Kawsar

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Correct Answer - A::B::C
We have, `|A(theta)|=1`
Hence, A is invertiable.
`A(pi+theta)=A(pi + theta)=[(sin(pi+theta),i cos (pi+theta)),(i cos (pi+theta),sin (pi+theta))]`
`=[(-sin theta,-i co theta),(-i cos theta,-sin theta)]=-A(theta)`
adj `(A(theta))=[(sin theta,-i cos theta),(-i cos theta,sin theta)]`
`implies A(theta)^(-1) =[(sin theta,-i cos theta),(-i cos theta,sin theta)]=A(pi-theta)`

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