Let \(A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{bmatrix}\)
and \(B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 1 \\ -1 & 1 & 0\end{bmatrix}\).
Both \(A\) and \(B\) can be transformed to the same matrix in
reduced row-echelon form using elementary row operations.
The answer is “True”.
Using Gauss-Jordan elimination, both matrices can be transformed to
\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\).
To transform \(A\) to this matrix, apply the following operations
in the order given:
\(R_1 \leftrightarrow R_3\), \(R_2 \leftrightarrow R_3\).
To transform \(B\), apply the following operations in the order given:
\(R_3 \leftarrow R_3 + R_1\), \(R_2 \leftarrow R_2 + (-3)R_3\),
\(R_2 \leftrightarrow R_3\).