Let A be a non-zero square matrix such that \(A' = kA\) for some real number k. Then the set of all possible values of k is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\{1, -1\}\)
Taking the transpose of both sides of \(A' = kA\) and using \((A')' = A\) and \((kA)' = kA'\):
\(A = kA'\)
\(A = k(kA) = k^2A\)
\((k^2 - 1)A = O\)
Since A is non-zero, \(k^2 - 1 = 0\), so \(k = 1\) or \(k = -1\).
Both values occur: \(k = 1\) for any non-zero symmetric matrix (such as I), and \(k = -1\) for any non-zero skew-symmetric matrix (such as \(\begin{bmatrix}0&1\\-1&0\end{bmatrix}\)). Hence the set is \(\{1, -1\}\).