If […] is a singular matrix, then the relation between a and b is:

The correct answer is Option (3) → $a+b=3$

Given

Matrix:

$A = \begin{bmatrix}-1 & 1 & 0 \\ a & b & 1 \\ 1 & 2 & 1\end{bmatrix}$

For a singular matrix, $|A| = 0$

Compute determinant:

$|A| = -1 \begin{vmatrix}b & 1 \\ 2 & 1\end{vmatrix} - 1 \begin{vmatrix}a & 1 \\ 1 & 1\end{vmatrix} + 0 \begin{vmatrix}a & b \\ 1 & 2\end{vmatrix}$

$= -1 (b\cdot1 - 1\cdot2) -1 (a\cdot1 - 1\cdot1) + 0 = -(b-2) - (a-1) = -b +2 -a +1 = -(a+b) +3$

Set $|A| = 0$:

$-(a+b) +3 =0 \Rightarrow a+b =3$