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Mathematics Question on Invertible Matrices

Which of the following matrices can NOT be obtained from the matrix [−12 1−1]\begin{bmatrix} -1 &2 \\\ 1 & -1 \end{bmatrix} by a single elementary row operation?

A

[01\1−1]\begin{bmatrix}0 & 1\\\1 & -1\end{bmatrix}

B

[1−1−12]\begin{bmatrix}1 &-1 \\\\-1 & 2\end{bmatrix}

C

[−12−27]\begin{bmatrix}-1 &2 \\\\-2 & 7\end{bmatrix}

D

[−12−13]\begin{bmatrix}-1 & 2\\\\-1 &3\end{bmatrix}

Answer

[−12−27]\begin{bmatrix}-1 &2 \\\\-2 & 7\end{bmatrix}

Explanation

Solution

(1) By R1→R1+R2, [01\1−1]\begin{bmatrix}0 &1\\\1 &-1\end{bmatrix} is possible

(2) By R1↔R2, [1−1−12]\begin{bmatrix}1&-1\\\\-1&2\end{bmatrix} is possible

(3) This matrix can’t be obtained
(4) By R2→R2+2R1, [−12−13]\begin{bmatrix}-1&2\\\\-1&3\end{bmatrix}is possible

So, the correct option is (C): [−12−27]\begin{bmatrix}-1&2\\\\-2&7\end{bmatrix}