If \( A=\begin{bmatrix} a & 1 & c\\ b & 2 & 1\\ 2 & 1 & c\end{bmatrix} \) is a singular matrix, then
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( a=2,c=\displaystyle\frac{1}{2},b\in\mathrm{R} \) (Set of real numbers)
A singular matrix has determinant 0. Expanding along the first row:
\( \displaystyle |A|=a(2c-1)-1(bc-2)+c(b-4) \)
\( \displaystyle =2ac-a-bc+2+bc-4c \)
\( \displaystyle =2ac-a-4c+2 \)
\( \displaystyle =a(2c-1)-2(2c-1) \)
\( \displaystyle =(a-2)(2c-1) \)
So A is singular exactly when a = 2 or \( c=\frac{1}{2} \), for any value of b (b cancels out). A condition on b alone, as in Options (3) and (4), cannot make A singular, and a = 2c does not make either factor 0 in general.
With a = 2 and \( c=\frac{1}{2} \) both factors are 0, so |A| = 0 for every real b. This is Option (2).