Question

If $\left[\begin{array}{c}\cos \theta -\sin \theta 0\\ \sin \theta \cos \theta 0\\ 001\end{array}\right]$, then $adjA=$

• A. $A^T$
• B. $1$
• C. $0$
• D. $A^2$

Answer 1

Answer: (A)

$A^T$

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

lets write co factor

$C_{11}+\cos \theta$ $C_{12}=-\sin \theta$ $C_{13}=0$

$C_{21}+\sin \theta$ $C_{22}=\cos \theta$ $C_{23}=0$

$C_{31}=0$ $C_{32}=0$ $C_{33}=1$

$\therefore$ co factor matrix

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

$\therefore AdjA=$ transpose of cofactor matrix

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

$=A^T$