Question

Which of the following matrices can NOT be obtained from the matrix \(\begin{bmatrix} -1 &2 \\  1 & -1 \end{bmatrix}\) by a single elementary row operation?

• A. \(\begin{bmatrix}0 & 1\\1 & -1\end{bmatrix}\)
• B. \(\begin{bmatrix}1 &-1 \\-1 & 2\end{bmatrix}\)
• C. \(\begin{bmatrix}-1 &2 \\-2 & 7\end{bmatrix}\)
• D. \(\begin{bmatrix}-1 & 2\\-1 &3\end{bmatrix}\)

Answer 1

The Correct Option is C

  To determine which matrix cannot be obtained from the given matrix by a single elementary row operation, we need to understand what elementary row operations are. They include:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a multiple of one row to another row.

Let's analyze each option:

1. \(\begin{bmatrix} 0 & 1 \\ 1 & -1 \end{bmatrix}\): This matrix can be obtained by adding Row 1 to Row 2 in the original matrix:

 

• Operation: \(R_1 \rightarrow R_1 + R_2\)
• \(\begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix} \longrightarrow \begin{bmatrix} 0 & 1 \\ 1 & -1 \end{bmatrix}\)

1. \(\begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}\): This matrix can be obtained by swapping Row 1 and Row 2:

 

• Operation: Swap \(R_1\) and \(R_2\)
• \(\begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix} \longrightarrow \begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}\)

1. \(\begin{bmatrix} -1 & 2 \\ -2 & 7 \end{bmatrix}\): This matrix does not result from any single elementary row operation:

 

• You cannot obtain Row 2 as \([-2, 7]\) by a single operation from the original Row 2, \([1, -1]\).

1. \(\begin{bmatrix} -1 & 2 \\ -1 & 3 \end{bmatrix}\): This matrix can be obtained by adding Row 1 to twice Row 2:

 

• Operation: \(R_2 \rightarrow R_2 + 2R_1\)
• \(\begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix} \longrightarrow \begin{bmatrix} -1 & 2 \\ -1 & 3 \end{bmatrix}\)

The matrix that cannot be obtained from the original matrix by a single elementary row operation is \(\begin{bmatrix} -1 & 2 \\ -2 & 7 \end{bmatrix}\).