Question

The values of $K$ for which the matrix

$A=\left[\begin{array}{ccc}1& 0& -K\\ 2& 1& 3\\ K& 0& 1\end{array}\right]$ is invertible, is:

• A. All the real values of $K$
• B. $K\in [-1,1]$
• C. No value of $K$
• D. $K\in \{-1,1\}$

Answer 1

The correct option is A All the real values of $K$

Matrix $A$ is invertible if $|A|\ne 0$

$i.e\left|\begin{array}{ccc}1& 0& -K\\ 2& 1& 3\\ K& 0& 1\end{array}\right|\ne 0$

$\Rightarrow 1\left(1\right)-K(-K)\ne 0$

$\Rightarrow |A|=K^2+1\ne 0$ which is true for all real values of $K$ .
Hence, $A$ is invertible for all real values of $K$.