Question

Matrix $A = \begin{bmatrix} 1 & 0 & - K \\ 2 & 1 & 3 \\ K & 0 & 1 \end{bmatrix}$is invertible for

• A. $K = 1$
• B. $K = - 1$
• C. $K = 0$
• D. All real K

Answer 1

Answer: (D)

All real K

Answer 2

For invertible, $|A| \neq 0$ i.e., $\left| \begin{matrix} 1 & 0 & - K \\ 2 & 1 & 3 \\ K & 0 & 1 \end{matrix} \right| \neq 0$

⇒ $1(1) - K( - K) \neq 0$ ⇒$|A| = K^{2} + 1 \neq 0$, which is true for all real K .