Let \(A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\). What is the least value of \(\theta\) for which \(A + A^{T} = I\), where \(I\) is the identity matrix of order 2?
\(\pi/3\)
\(A + A^{T} = \begin{pmatrix} 2\cos\theta & 0 \\ 0 & 2\cos\theta \end{pmatrix} = 2\cos\theta \, I\). Setting this equal to \(I\) gives \(2\cos\theta = 1\), so \(\cos\theta = 1/2\) and the least positive value is \(\theta = \pi/3\).
Similar Questions
If \[ A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \] then what is \( AA^{T} \) equal to (where \( A^{T} \) is the transpose of \( A \))?
If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?
Important Questions from Transpose of a Matrix
If $A$ and $B$ are two symmetric matrices of the same order, then $AB - BA$ is a:
If \[ A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \] then what is \( AA^{T} \) equal to (where \( A^{T} \) is the transpose of \( A \))?
If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?
