Question

If A is a square matrix satisfying A^T A = I, write the value of |A|.

Answer 1

\[\text{ Let }A = \left[ a_{i j} \right]\text{ be a square matrix of order n .} \] 
Here, 
\[\left| A \right| = \left| A^T \right| \left[\text{ By property of determinants }\right]\] 
\[\text{Given:} A^T A = I \] 
\[ \Rightarrow \left| A^T A \right| = 1\] 
Then,
\[\left| A^T A \right| = \left| A^T \right| \left| A \right| \left[\text{ Since the determinants are of the same order }\right] \] 
\[ \Rightarrow \left| A^T \right| \left| A \right| = 1\] 
\[ \Rightarrow \left| A \right| = \frac{1}{\left| A^T \right|}\] 
\[ \Rightarrow \left| A \right| = \frac{1}{\left| A \right|} \left[ \therefore \left| A \right| = \left| A^T \right| \right]\] 
\[ \Rightarrow \left| A \right|^2 = 1\]
\[ \Rightarrow \left| A \right| = \pm 1\]