Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

I₃ = 1 0 0 0 1 0 0 0 1

$\text{elementary ROW transformations only}$

$\text{To find: which option is reachable from } I_3 \text{ by row ops}$

$\text{row op} \Leftrightarrow \text{left-multiply by an invertible } E$

$\Rightarrow \text{ rank preserved, } \det \ne 0 \text{ preserved}$

$A \text{ reachable from } I_3 \iff A \text{ invertible} \iff \det A \ne 0$ $1.\ \det \text{ of each option}$ $2.\ \text{pick the one with } \det \ne 0$ $\det A_1 = 0 \quad (\text{all three rows identical})$ $\det A_2 = 1(3-8) - 1(2-4) + 1(4-3) = -5+2+1 = -2 \ne 0$ $A_3:\ R_3 = 3R_2 - 4R_1 \Rightarrow \det A_3 = 0$ $A_4:\ R_3 = R_1 + R_2 \Rightarrow \det A_4 = 0$ $\text{only } \det A_2 = -2 \ne 0 \Rightarrow A_2 \text{ is reachable}$ $\boxed{\text{option (2)}}$