If f- - cos- -sin- 0 sin- cos- 0 0 0 1 - find f-

f(α)=[ cosα amp; sinα amp; 0; sinα amp; cosα amp; 0; 0 amp; 0 amp; 1 ] f( α)=[ cos( α) amp; sin( α) ...

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Equality of Matrices

If fα =[ co...

Question

If $f (α) = ⎡ ⎢ ⎣ \begin{matrix} cos α & - sin α & 0 sin α & cos α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$ find
$f (- α)$

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F (α) = [\begin{matrix} \cos α & - \sin α & 0 \\ \sin α & \cos α & 0 \\ 0 & 0 & 1 \end{matrix}] and G (β) = [\begin{matrix} \cos β & 0 & \sin β \\ 0 & 1 & 0 \\ - \sin β & 0 & \cos β \end{matrix}]

{[F (α)]}^{- 1} = F (- α)

{[G (β)]}^{- 1} = G (- β)

{[F (α) G (β)]}^{- 1} = G (- β) F (- α)

(α)

⎡ ⎢ ⎢ ⎣ \begin{matrix} cos α & - sin α & 0 sin α & cos α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

α ϵ R

F (α) . F (- α) = I_{3}

[F (α)]^{- 1} = F (- α)

F (α) = ⎡ ⎢ ⎣ \begin{matrix} c o s α & - s i n α & 0 s i n α & c o s α & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

[F (α)]^{- 1} = F (- α)

G (β) = ⎡ ⎢ ⎣ \begin{matrix} c o s β & 0 & s i n β 0 & 1 & 0 - s i n β & 0 & c o s β \end{matrix} ⎤ ⎥ ⎦

[G (β)]^{- 1} = G (- β)

∣ ∣ ∣ ∣ \begin{matrix} cos α & - sin α & 1 sin α & cos α & 1 c o s (α + β) & - sin (α + β) & 1 \end{matrix} ∣ ∣ ∣ ∣

α

f (α) = λ,

f (α)

α

F (α) = ⎛ ⎜ ⎝ \begin{matrix} cos α & - sin α & 0 sin α & cos α & 0 0 & 0 & 1 \end{matrix} ⎞ ⎟ ⎠

{F (x)}^{- 1} = ?

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