Question

Find the adjoint and the inverse of the matrix \left[\begin{array}{ccc}cos\alpha&-sin\alpha\\sin\alpha&cos\alpha\end{array}\right]

Answer 1

Answer:

$adj.A=\left[\begin{array}{ccc}cos\alpha&sin\alpha\\-sin\alpha&cos\alpha\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}cos\alpha&sin\alpha\\\\-sin\alpha&cos\alpha\end{array}\right]$

Step-by-step explanation:

As we know that Adjoint of matrix is calculated as Minor × Co-factor of each element and taking transpose of it.

or

$adj.A=[A_{ji}]_{n\times n}$

$A= \left[\begin{array}{ccc}cos\alpha&-sin\alpha\\sin\alpha&cos\alpha\end{array}\right]\\\\adj.A=\left[\begin{array}{ccc}cos\alpha&-sin\alpha\\sin\alpha&cos\alpha\end{array}\right]^{'}\\\\adj.A=\left[\begin{array}{ccc}cos\alpha&sin\alpha\\-sin\alpha&cos\alpha\end{array}\right]$

Now

$A^{-1} =\frac{adj.A}{|A|} \\\\|A|=1\\\\\\A^{-1} =\left[\begin{array}{ccc}cos\alpha&sin\alpha\\-sin\alpha&cos\alpha\end{array}\right]$