78824
If \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) and \(A \cdot \operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), then the value of \(k\) is
1 \(\sin \mathrm{x} \cos \mathrm{x} \quad\)
2 1
3 2
4 8
Explanation:
(B) : Given that, \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) \(\operatorname{Adj}(A)=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) \(A \cdot \operatorname{Adj} A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) So, A. Adj \(A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \quad\) (given) \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} \mathrm{k} & 0 \\ 0 & \mathrm{k} \end{array}\right]\) On comparing corresponding elements on the both side we get - \(\mathrm{k}=1\)
SRM JEE-2011
Matrix and Determinant
78826
If \(A=\left[\begin{array}{cc}1 & -2 \\ 3 & 0\end{array}\right], B=\left[\begin{array}{cc}-1 & 4 \\ 2 & 3\end{array}\right]\) and \(C=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\) then \(5 \mathrm{~A}-3 \mathrm{~B}+2 \mathrm{C}\) is equal to
78824
If \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) and \(A \cdot \operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), then the value of \(k\) is
1 \(\sin \mathrm{x} \cos \mathrm{x} \quad\)
2 1
3 2
4 8
Explanation:
(B) : Given that, \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) \(\operatorname{Adj}(A)=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) \(A \cdot \operatorname{Adj} A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) So, A. Adj \(A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \quad\) (given) \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} \mathrm{k} & 0 \\ 0 & \mathrm{k} \end{array}\right]\) On comparing corresponding elements on the both side we get - \(\mathrm{k}=1\)
SRM JEE-2011
Matrix and Determinant
78826
If \(A=\left[\begin{array}{cc}1 & -2 \\ 3 & 0\end{array}\right], B=\left[\begin{array}{cc}-1 & 4 \\ 2 & 3\end{array}\right]\) and \(C=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\) then \(5 \mathrm{~A}-3 \mathrm{~B}+2 \mathrm{C}\) is equal to
78824
If \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) and \(A \cdot \operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), then the value of \(k\) is
1 \(\sin \mathrm{x} \cos \mathrm{x} \quad\)
2 1
3 2
4 8
Explanation:
(B) : Given that, \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) \(\operatorname{Adj}(A)=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) \(A \cdot \operatorname{Adj} A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) So, A. Adj \(A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \quad\) (given) \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} \mathrm{k} & 0 \\ 0 & \mathrm{k} \end{array}\right]\) On comparing corresponding elements on the both side we get - \(\mathrm{k}=1\)
SRM JEE-2011
Matrix and Determinant
78826
If \(A=\left[\begin{array}{cc}1 & -2 \\ 3 & 0\end{array}\right], B=\left[\begin{array}{cc}-1 & 4 \\ 2 & 3\end{array}\right]\) and \(C=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\) then \(5 \mathrm{~A}-3 \mathrm{~B}+2 \mathrm{C}\) is equal to
78824
If \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) and \(A \cdot \operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), then the value of \(k\) is
1 \(\sin \mathrm{x} \cos \mathrm{x} \quad\)
2 1
3 2
4 8
Explanation:
(B) : Given that, \(A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) \(\operatorname{Adj}(A)=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) \(A \cdot \operatorname{Adj} A=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\) A. \(\operatorname{Adj} A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) So, A. Adj \(A=k\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \quad\) (given) \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{ll} \mathrm{k} & 0 \\ 0 & \mathrm{k} \end{array}\right]\) On comparing corresponding elements on the both side we get - \(\mathrm{k}=1\)
SRM JEE-2011
Matrix and Determinant
78826
If \(A=\left[\begin{array}{cc}1 & -2 \\ 3 & 0\end{array}\right], B=\left[\begin{array}{cc}-1 & 4 \\ 2 & 3\end{array}\right]\) and \(C=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]\) then \(5 \mathrm{~A}-3 \mathrm{~B}+2 \mathrm{C}\) is equal to
