Matrices and Determinants

Concept:Any matrix obtained by elementary row transformations on the identity matrix is invertible (non-zero determinant).Explanation:Elementary row operations correspond to multiplying the identity matrix by elementary matrices (each with determinant $1$, $-1$, or a non‑zero constant).Hence the resulting matrix has a non‑zero determinant.Check the determinant of each option:(A) $\det\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}=0$(B) $\det\begin{bmatrix}1&1&1\\2&3&4\\1&2&1\end{bmatrix}=-2\neq0$(C) $\det\begin{bmatrix}1&1&1\\2&3&4\\2&5&8\end{bmatrix}=0$(D) $\det\begin{bmatrix}1&1&1\\-1&1&2\\0&2&3\end{bmatrix}=0$Only option (B) has a non‑zero determinant, so it can be obtained from the identity matrix by elementary row transformations.An explicit sequence starting from $I=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$ is:$R_2\to R_2+2R_3$, $R_1\to R_1-R_3$, $R_3\to -2R_3$, $R_3\to R_3+R_2$, $R_1\to R_1+R_2$, $R_3\to R_3+R_1$, $R_2\to R_2+2R_1$  →  $\begin{bmatrix}1&1&1\\2&3&4\\1&2&1\end{bmatrix}$.Answer:Option B.