78637 If \(A\) and \(B\) are two matrices such that \(A B=B\) and \(B A=A\) then \(A^{2}+B^{2}=\)

78638 \(G=\left\{\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right): \theta \in \mathbf{R}\right\}\) is a group under matrix multiplication. Then which one of the following statements in respect of \(G\) is true.

78639 If \(A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]\) then \((A B)^{T}\) is equal to

78641 If \(\left(x_{1}, y_{1}\right),\left(y_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) are the vertices of a triangle whose area is \(k\) square units, then
\(\left|\begin{array}{lll}
\mathbf{x}_{1} & \mathbf{y}_{1} & 4 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 4 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 4 \end{array}\right|^{2} \text { is }\)

78642 If \(2\left[\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right]+\left[\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 8\end{array}\right]\) then the value of \(x\)

78637 If \(A\) and \(B\) are two matrices such that \(A B=B\) and \(B A=A\) then \(A^{2}+B^{2}=\)

78638 \(G=\left\{\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right): \theta \in \mathbf{R}\right\}\) is a group under matrix multiplication. Then which one of the following statements in respect of \(G\) is true.

78639 If \(A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]\) then \((A B)^{T}\) is equal to

78641 If \(\left(x_{1}, y_{1}\right),\left(y_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) are the vertices of a triangle whose area is \(k\) square units, then
\(\left|\begin{array}{lll}
\mathbf{x}_{1} & \mathbf{y}_{1} & 4 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 4 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 4 \end{array}\right|^{2} \text { is }\)

78642 If \(2\left[\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right]+\left[\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 8\end{array}\right]\) then the value of \(x\)

78637 If \(A\) and \(B\) are two matrices such that \(A B=B\) and \(B A=A\) then \(A^{2}+B^{2}=\)

78638 \(G=\left\{\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right): \theta \in \mathbf{R}\right\}\) is a group under matrix multiplication. Then which one of the following statements in respect of \(G\) is true.

78639 If \(A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]\) then \((A B)^{T}\) is equal to

78641 If \(\left(x_{1}, y_{1}\right),\left(y_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) are the vertices of a triangle whose area is \(k\) square units, then
\(\left|\begin{array}{lll}
\mathbf{x}_{1} & \mathbf{y}_{1} & 4 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 4 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 4 \end{array}\right|^{2} \text { is }\)

78642 If \(2\left[\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right]+\left[\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 8\end{array}\right]\) then the value of \(x\)

78637 If \(A\) and \(B\) are two matrices such that \(A B=B\) and \(B A=A\) then \(A^{2}+B^{2}=\)

78638 \(G=\left\{\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right): \theta \in \mathbf{R}\right\}\) is a group under matrix multiplication. Then which one of the following statements in respect of \(G\) is true.

78639 If \(A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]\) then \((A B)^{T}\) is equal to

78641 If \(\left(x_{1}, y_{1}\right),\left(y_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) are the vertices of a triangle whose area is \(k\) square units, then
\(\left|\begin{array}{lll}
\mathbf{x}_{1} & \mathbf{y}_{1} & 4 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 4 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 4 \end{array}\right|^{2} \text { is }\)

78642 If \(2\left[\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right]+\left[\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 8\end{array}\right]\) then the value of \(x\)

78637 If \(A\) and \(B\) are two matrices such that \(A B=B\) and \(B A=A\) then \(A^{2}+B^{2}=\)

78638 \(G=\left\{\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right): \theta \in \mathbf{R}\right\}\) is a group under matrix multiplication. Then which one of the following statements in respect of \(G\) is true.

78639 If \(A=\left[\begin{array}{ccc}1 & -2 & 1 \\ 2 & 1 & 3\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 1\end{array}\right]\) then \((A B)^{T}\) is equal to

78641 If \(\left(x_{1}, y_{1}\right),\left(y_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) are the vertices of a triangle whose area is \(k\) square units, then
\(\left|\begin{array}{lll}
\mathbf{x}_{1} & \mathbf{y}_{1} & 4 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 4 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 4 \end{array}\right|^{2} \text { is }\)

78642 If \(2\left[\begin{array}{ll}1 & 3 \\ 0 & x\end{array}\right]+\left[\begin{array}{ll}y & 0 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}5 & 6 \\ 1 & 8\end{array}\right]\) then the value of \(x\)