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 \( A \))?
Identity matrix
We have \[ A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \]
Now, \[ AA^T = \begin{bmatrix} \cos^2 \alpha + \sin^2 \alpha & 0 \\ 0 & \cos^2 \alpha + \sin^2 \alpha \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \]
Similar Questions
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?
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?
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 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?
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?
