Question

Which one of the following matrices can be obtained by performing elementary row transformations on the \(3\times3\) identity matrix?

• A. \[ \begin{bmatrix} 1&1&1 1&1&1 1&1&1 \end{bmatrix} \]
• B. \[ \begin{bmatrix} 1&1&1 2&3&4 1&2&1 \end{bmatrix} \]
• C. \[ \begin{bmatrix} 1&1&1 2&3&4 2&5&8 \end{bmatrix} \]
• D. \[ \begin{bmatrix} 1&1&1 -1&1&2 0&2&3 \end{bmatrix} \]

Hint:

Matrices obtained from elementary row operations on identity matrix are always: \[ \mathrm{Invertible} \] Hence: \[ \det(A)\neq0 \]

Answer 1

The Correct Option is B

Step 1: Use the property of elementary row transformations.
A matrix obtained from identity matrix using elementary row operations must be: \[ \mathrm{Non\text{-}singular} \] Thus determinant must satisfy: \[ \det(A)\ne 0 \]

Step 2: Check Option (A).
\[ \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix}=0 \] All rows are identical. \[ \Rightarrow \mathrm{Singular} \] Therefore: \[ \Rightarrow \mathrm{Option\ (A)\ is\ Incorrect} \]

Step 3: Check Option (B).
\[ \begin{vmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 2 & 1 \end{vmatrix} \] Expanding: \[ =1(3-8)-1(2-4)+1(4-3) \] \[ =-5+2+1 \] \[ =-2 \] Since determinant is non-zero: \[ \Rightarrow \mathrm{Non\text{-}singular} \] Thus: \[ \Rightarrow \mathrm{Option\ (B)\ can\ be\ obtained} \]

Step 4: Check Option (C).
\[ \begin{vmatrix} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 2 & 5 & 8 \end{vmatrix} \] Expanding: \[ =1(24-20)-1(16-8)+1(10-6) \] \[ =4-8+4 \] \[ =0 \] Therefore: \[ \Rightarrow \mathrm{Option\ (C)\ is\ Incorrect} \]

Step 5: Check Option (D).
\[ \begin{vmatrix} 1 & 1 & 1 \\ -1 & 1 & 2 \\ 0 & 2 & 3 \end{vmatrix} \] Expanding: \[ =1(3-4)-1(-3)+1(-2) \] \[ =-1+3-2 \] \[ =0 \] Thus: \[ \Rightarrow \mathrm{Option\ (D)\ is\ Singular} \]

Step 6: Identify the correct option.
Only option \[ \boxed{\mathrm{(B)}} \] can be obtained from elementary row transformations on identity matrix.