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Question
If A is a square matrix of order 3 such that adj (2A) = k adj (A), then write the value of k.
Solution
\[\text{ For any matrtix A of order n, adj }\left( \lambda A \right) = \lambda^{n - 1} \left( adj A \right), \text{ where }\lambda \text{ is a constant .} \]
Thus, for matrix A of order 3, we have
\[ adj (2A) = 2^{3 - 1} \left( adj A \right)\]
\[ \Rightarrow adj (2A) = 2^2 \left( adj A \right)\]
\[ \Rightarrow adj (2A) = 4 adj \left( A \right)\]
\[ \Rightarrow kadj (A) = 4 adj \left( A \right) \left[ \because adj (2A) = k adj \left( A \right) \right]\]
\[ \Rightarrow k = 4\]
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