Question

Let A be a 3 × 3 square matrix, such that A (adj A) = 2 I, where I is the identity matrix. Write the value of |adj A|.

Answer 1

\[\text{ We know that for a matix A of order n, A . }\left( adj A \right) = \left| A \right| I_n ,\text{ where I is the identity matrix .} \]
\[\text{ Given: }A . \left( adj A \right) = 2I\]
\[ \Rightarrow \left| A \right|I = 2I\]
\[ \Rightarrow \left| A \right| = 2\]
Now,
\[\left| adj A \right| = \left| A \right|^{n - 1} \]
\[ \Rightarrow \left| adj A \right| = 2^2 = 4\]