Question

On a matrix A when three elementary operations namely interchange of \( R_1 \) and \( R_2 \), \( R_2 \rightarrow R_2 - 2R_1 \), \( R_3 \rightarrow R_3 - 3R_1 \) are applied successively, A is transformed to \( \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 5 \\ 6 & 1 & 2 \end{bmatrix} \). Then \( \text{Tr}(A) = \)

• A. \( 12 \)
• B. \( 21 \)
• C. \( 4 \)
• D. \( 20 \)

Hint:

The trace of a matrix is invariant under cyclic permutation of matrices, but not under arbitrary elementary row operations. In this context, evaluate the trace directly from the provided final state.

Answer 1

The Correct Option is C

Step 1: Read what the question really wants.
This problem asks for the trace, which is just the sum of the diagonal entries, of the displayed transformed matrix. So we focus on its main diagonal.
Step 2: Write out the matrix.
The transformed matrix is \[ \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 5 \\ 6 & 1 & 2 \end{bmatrix}. \]
Step 3: Pick out the diagonal entries.
The main diagonal runs top-left to bottom-right: the $(1,1)$ entry is $1$, the $(2,2)$ entry is $1$, and the $(3,3)$ entry is $2$.
Step 4: Recall the trace definition.
The trace is simply $a_{11} + a_{22} + a_{33}$, nothing more.
Step 5: Add them up.
\[ 1 + 1 + 2 = 4. \]
Step 6: State the answer.
So the required trace is $4$, which is option (C).
\[ \boxed{4} \]