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: Understanding the Concept:
Elementary row transformations (ERTs) performed on a matrix preserve its rank and the property of being non-singular. If we start with the identity matrix \( I \), which has a determinant of \( 1 \) (non-zero), any matrix obtained from it via ERTs must also have a non-zero determinant. Conversely, any non-singular (invertible) matrix can be transformed into the identity matrix using Gaussian elimination.
Step 2: Key Formula or Approach:
We evaluate the determinant \( \Delta \) of each matrix. The matrix obtained from \( I \) will be the one for which \( \Delta \neq 0 \).
Step 3: Detailed Explanation:
1. Matrix (A):
\[ \Delta = \begin{vmatrix} 1 & 1 & 1
1 & 1 & 1
1 & 1 & 1 \end{vmatrix} \]
All rows are identical. Therefore, \( \text{rank} = 1 \) and \( \Delta = 0 \). This matrix is singular and cannot be obtained from \( I \).
2. Matrix (B):
\[ \Delta = 1(3 \cdot 1 - 2 \cdot 4) - 1(2 \cdot 1 - 1 \cdot 4) + 1(2 \cdot 2 - 1 \cdot 3) \]
\[ \Delta = 1(3 - 8) - 1(2 - 4) + 1(4 - 3) = -5 + 2 + 1 = -2 \]
Since \( \Delta = -2 \neq 0 \), this matrix is non-singular. It can be obtained from \( I \) through row operations.
3. Matrix (C):
\[ \Delta = 1(3 \cdot 8 - 5 \cdot 4) - 1(2 \cdot 8 - 2 \cdot 4) + 1(2 \cdot 5 - 2 \cdot 3) \]
\[ \Delta = 1(24 - 20) - 1(16 - 8) + 1(10 - 6) = 4 - 8 + 4 = 0 \]
The matrix is singular (\( R_3 = 2R_2 - 2R_1 \)). Cannot be obtained from \( I \).
4. Matrix (D):
\[ \Delta = 1(1 \cdot 3 - 2 \cdot 2) - 1(-1 \cdot 3 - 0) + 1(-1 \cdot 2 - 0) \]
\[ \Delta = 1(3 - 4) - 1(-3) + 1(-2) = -1 + 3 - 2 = 0 \]
The matrix is singular (\( R_2 = R_3 - R_1 \)). Cannot be obtained from \( I \).
Step 4: Final Answer:
Only matrix (B) is invertible.