If $\begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix}$ is square root of identity matrix of order $2$ then
- $ 1 + \alpha^2 + \beta \gamma = 0$
- $ 1 + \alpha^2 - \beta \gamma = 0$
- $ 1 - \alpha^2 + \beta \gamma = 0$
- $\alpha^2 + \beta \gamma = 1$
The Correct Option is D
Solution and Explanation
$ \begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix} \begin{bmatrix}\alpha &\beta \\ \gamma &-\alpha \end{bmatrix} = \begin{bmatrix}1&0\\ 0&1\end{bmatrix} $
$\Rightarrow \alpha^{2} + \beta\gamma = 1 $
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Concepts Used:
Transpose of a Matrix
The matrix acquired by interchanging the rows and columns of the parent matrix is called the Transpose matrix. The transpose matrix is also defined as - “A Matrix which is formed by transposing all the rows of a given matrix into columns and vice-versa.”
The transpose matrix of A is represented by A’. It can be better understood by the given example:
Now, in Matrix A, the number of rows was 4 and the number of columns was 3 but, on taking the transpose of A we acquired A’ having 3 rows and 4 columns. Consequently, the vertical Matrix gets converted into Horizontal Matrix.
Hence, we can say if the matrix before transposing was a vertical matrix, it will be transposed to a horizontal matrix and vice-versa.
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