Vector Algebra ...

If \(A=\left[\begin{array}{cc}\cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta\end{array}\right]\) and \(A + A ^{T}= I\) Where \(I\) is the unit matrix of \(2 \times 2\) and  \(A ^{T}\) is the transpose of \(A\), then the value of \(\theta\) is equal to:

Consider the following statements in respect of the matrix \(A=\left[\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right]\)

1) The matrix \(A\) is skew-symmetric.

2) The matrix \(A\) is symmetric.

3) The matrix \(A\) is Iinvertible.

Which of the above statements is/are correct?

If the matrix \(\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]\) is singular, then \(\theta=?\)

If \(\left[\begin{array}{ccc}1 & -4 & 3 \\ 0 & 6 & -7 \\ 2 & 4 & \lambda\end{array}\right]\) is not an invertible matrix, then what is the value of \(\lambda\)?

If \(x+2 y=\left[\begin{array}{cc}2 & -3 \\ 1 & 5\end{array}\right]\) and \(2 x+5 y=\left[\begin{array}{ll}7 & 5 \\ 2 & 3\end{array}\right]\), then \(y\) is equal to:

If \(A=\left[\begin{array}{ccc}1 & 3+x & 2 \\ 1-x & 2 & y+1 \\ 2 & 5-y & 3\end{array}\right]\) is a symmetric matrix, then \(3 x+y\) is equal to:

If \(A =\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]\) then \(AA ^{T}\) is equal to: (where \(A ^{ T }\) is the transpose of \(\left.A \right)\)

Find \(2 X-Y\) matrix such as \(X+Y=\left[\begin{array}{ll}7 & 5 \\ 3 & 4\end{array}\right]\) and \(X-Y=\left[\begin{array}{cc}1 & -3 \\ 3 & 0\end{array}\right]\).

If \(3 A +4 B ^{\prime}=\left[\begin{array}{ccc}7 & -10 & 17 \\ 0 & 6 & 31\end{array}\right]\) and \(2 B -3 A ^{\prime}=\left[\begin{array}{rr}-1 & 18 \\ 4 & 0 \\ -5 & -7\end{array}\right]\) then \(B =\) ?