JEE Advanced 2026 Paper 1, Mathematics Section 1 Q3: Algebra Of Matrices

A matrix is obtainable from the identity matrix by a sequence of elementary row operations if and only if it is non-singular (invertible), i.e. has a non-zero determinant.

Evaluating the four determinants:

(A) $\left|\begin{array}{c}111\\ 111\\ 111\end{array}\right|=0$ (identical rows).

(B) $\left|\begin{array}{c}111\\ 234\\ 121\end{array}\right|=1(3-8)-1(2-4)+1(4-3)=-5+2+1=-2\ne 0$.

(C) $\left|\begin{array}{c}111\\ 234\\ 258\end{array}\right|=1(24-20)-1(16-8)+1(10-6)=4-8+4=0$.

(D) $\left|\begin{array}{c}111\\ -112\\ 023\end{array}\right|=1(3-4)-1(-3-0)+1(-2-0)=-1+3-2=0$.

Only matrix (B) is non-singular.

$\therefore$ The correct option is (B).