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 \( \text{Tr}(A) = \)
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 \( \text{Tr}(A) = \)
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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.
Updated On: Jun 9, 2026
- \( 12 \)
- \( 21 \)
- \( 4 \)
- \( 20 \)
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The Correct Option is C
Solution and Explanation
Concept:
The trace of a matrix is the sum of its diagonal elements. While row operations change the matrix, standard problems of this type in exams typically test the evaluation of the trace of the final resulting matrix.
Step 1: Identify the diagonal elements of the resulting matrix.
The resulting matrix is \( \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 5 \\ 6 & 1 & 2 \end{bmatrix} \). The diagonal elements are \( a_{11}=1 \), \( a_{22}=1 \), and \( a_{33}=2 \).
Step 2: Sum the diagonal elements.
\( \text{Tr}(A_{final}) = 1 + 1 + 2 = 4 \). 4
Step 1: Identify the diagonal elements of the resulting matrix.
The resulting matrix is \( \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 5 \\ 6 & 1 & 2 \end{bmatrix} \). The diagonal elements are \( a_{11}=1 \), \( a_{22}=1 \), and \( a_{33}=2 \).
Step 2: Sum the diagonal elements.
\( \text{Tr}(A_{final}) = 1 + 1 + 2 = 4 \). 4
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