If \(A^{-1}\) exists for the matrix \(A = \begin{bmatrix}1&\lambda&-1\\-1&1&0\\ \lambda&1&1\end{bmatrix}\) then
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\lambda \ne -1\)
\(A^{-1}\) exists only when A is non-singular, that is, \(|A| \ne 0\). Expanding along the first row:
\(|A| = 1(1 - 0) - \lambda[(-1)(1) - 0] + (-1)[(-1)(1) - \lambda]\)
\(= 1 + \lambda + 1 + \lambda\)
\(= 2 + 2\lambda\)
\(2 + 2\lambda \ne 0\)
\(\lambda \ne -1\)