Correct option is (A) 3
Given,
\(A=\left[\begin{array}{ccc}
1 & -2 & \alpha \\
\alpha & 2 & -1
\end{array}\right]\)
\(\text { and } B=\left[\begin{array}{cc}
2 & \alpha \\
-1 & 2 \\
4 & -5
\end{array}\right]\)
\(A B=\left[\begin{array}{ccc}
1 & -2 & \alpha \\
\alpha & 2 & -1
\end{array}\right]\left[\begin{array}{cc}
2 & \alpha \\
-1 & 2 \\
4 & -5
\end{array}\right]\)
\(=\left[\begin{array}{cc}
4+4 \alpha & -4 \alpha-4 \\
2 \alpha-6 & \alpha^2+9
\end{array}\right]\)
Given,
\(|A B|=0\)
\(\therefore\left|\begin{array}{cc}
4+4 \alpha & -4 \alpha-4 \\
2 \alpha-6 & \alpha^2+9
\end{array}\right|=0\)
\( \Rightarrow(4 \alpha+4)\left|\begin{array}{cc}
1 & -1 \\
2 \alpha-6 & \alpha^2+9
\end{array}\right|=0\)
\(\Rightarrow(4 \alpha+4)\left(\alpha^2+9+2 \alpha-6\right)=0\)
\(\Rightarrow(4 \alpha+4)\left(\alpha^2+2 \alpha+3\right)=0\)
\(\therefore \alpha-=-1\)
\(\text {or } \alpha^2+2 \alpha+3=0\)
\( \alpha_1+\alpha_2=-2\)
\(\therefore\) Sum of all values of \(\alpha=-1-2=-3\)
\(\therefore\) Absolute value of \(\alpha=|-3|=3\)
