Question

If $\left[ {\begin{array}{*{20}{c}}
  \alpha &\beta \\
  \gamma &{ - \alpha }
\end{array}} \right]$ is to be the square root of two rowed unit matrices, then $\alpha ,\;\beta \;{\text{and}}\;\gamma $ should satisfy the relation.
A) $1 + {\alpha ^2} + \beta \gamma = 0$
B) $1 - {\alpha ^2} - \beta \gamma = 0$
C) $1 - {\alpha ^2} + \beta \gamma = 0$
D) $1 + {\alpha ^2} - \beta \gamma = 0$

Answer 1

Hint: The given matrix is a square root of two rowed unit matrix, and the square root of two rowed unit matrix is given as $ \pm \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]$. Both matrices are the square root of two rowed matrices, that is there each element should be equal, hence find the respective values of the given variables and put them in the options to find the correct one.

Complete step-by-step solution:
To find out which relation is being satisfied by the variables $\left( {\alpha ,\;\beta \;{\text{and}}\;\gamma } \right)$ present in the given matrix, we have to first find their respective values.
Since it is given in the question that the matrix $\left[ {\begin{array}{*{20}{c}}
  \alpha &\beta \\
  \gamma &{ - \alpha }
\end{array}} \right]$ is the square root of two rowed matrix, but we also know that the square root of a two rowed matrix is given as follows $ \pm \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]$
Here we get two matrices that are supposed to be the square root of two rowed matrix that means both of them are equal matrices, so equating both matrices element by element, we will get,
$
   \Rightarrow \left[ {\begin{array}{*{20}{c}}
  \alpha &\beta \\
  \gamma &{ - \alpha }
\end{array}} \right] = \pm \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right] \\
   \Rightarrow \alpha = \pm 1\;{\text{or}}\; \mp 1\;{\text{and}}\;\beta = \gamma = 0 \\
   \Rightarrow \alpha = \pm 1\;{\text{and}}\;\beta = \gamma = 0 \\
 $
Now on substituting their respective values in above options, we get that
Option B,
$
   = 1 - {\alpha ^2} - \beta \gamma \\
   = 1 - 1 - 0 \\
   = 0 \\
 $
And option C,
$
   = 1 - {\alpha ^2} + \beta \gamma \\
   = 1 - 1 + 0 \\
   = 0 \\
 $
Hence options value of $\alpha ,\;\beta \;{\text{and}}\;\gamma $ are satisfying option B and C, so option B and C both are the answer.

Hence the correct options are ‘B’ and ‘C’.

Note: Two or more matrices are said to be equal matrices when all of their elements are equal to one another matrix’s respective elements, that’s why we have equated the elements of square root of a two rowed matrix and the given matrix.