Question

If A = $\left[\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right]$, then A^-1 exists if
(a) λ=2 (b) λ≠2 (c) λ≠-2 (d) none of these

Answer 1

Given: A = $\left[\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right]$

A⁻¹ exists only if |A| ≠ 0.
$$\begin{gathered} \left|\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right|\ne 0 \\ \Rightarrow 2\left(6-5\right)+1\left(5\lambda +6\right)\ne 0 \\ \Rightarrow 2\left(1\right)+5\lambda +6\ne 0 \\ \Rightarrow 2+5\lambda +6\ne 0 \\ \Rightarrow 5\lambda +8\ne 0 \\ \Rightarrow 5\lambda \ne -8 \\ \Rightarrow \lambda \ne \frac{-8}{5} \end{gathered}$$

Thus, ​A⁻¹ exists if λ ∈ $R-\left\{-\frac{8}{5}\right\}$

Hence, the correct option is (a).