<s>[INST] Which of the following matrices has the same determinant as matrix $G$ Which of the following matrices has the  same determinant as matrix: 
$G$ =  $ \begin{bmatrix}
    1       & 2 & 3  \\
    -2       & -3 & 2 \\
    1       & 2 & 1 
\end{bmatrix}$
I could just calculate each individual determinant; however I am preparing for the CSET exam with no calculators and need to be efficient with my time.
Is there another way to identify the solution?
A.  $ \begin{bmatrix}
    -2       & -3 & 2  \\
    1       & 2 & 3\\
    1       & 2 & 1 
\end{bmatrix}$
B.  $ \begin{bmatrix}
    1       & 2 & 3  \\
    -2       & -3 & 2 \\
    2       & 4 & 4 
\end{bmatrix}$
C.  $ \begin{bmatrix}
    2       & 4 & 6  \\
    -2       & -3 & 2 \\
    1       & 2 & 1 
\end{bmatrix}$
D.  $ \begin{bmatrix}
    2       & 1 & 3  \\
    -3       & -2 & 2 \\
    2      & 1 & 1 
\end{bmatrix}$
 [/INST] Note that the matrix $B$ is obtained from $G$ by adding $\DeclareMathOperator{Row}{Row}\Row_1(G)$ to $\Row_3(G)$. Elementary row operations of this form leave the determinant unchanged. Thus $\det B=\det G$.
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