Question

Multiplication of matrices E and F is G matrices E and G are

E = $\left[\begin{array}{ccc}cos\theta & -sin\theta & 0\\ sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]$ and

G = $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

What is the matrix F?

• A. $\left[\begin{array}{ccc}cos\theta & -sin\theta & 0\\ sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}cos\theta & cos\theta & 0\\ -cos\theta & sin\theta & 0\\ 0& 0& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}cos\theta & sin\theta & 0\\ -sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]$
• D. $\left[\begin{array}{ccc}sin\theta & -cos\theta & 0\\ cos\theta & sin\theta & 0\\ 0& 0& 1\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{ccc}cos\theta & sin\theta & 0\\ -sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]$

Given,

E= $\left[\begin{array}{ccc}cos\theta & -sin\theta & 0\\ sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]$

G = $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ = I

and EF = G = I

$\Rightarrow$ F = $E^{-1}$

Now, $\left|\begin{array}{c}E\end{array}\right|$ = 1

$E^{-1}$ = $\left[\begin{array}{ccc}cos\theta & sin\theta & 0\\ -sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]$