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\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
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\begin{document}

{\bf Question}

Given that

$$\left|\begin{array} {ccc} 5 &amp; 4 &amp; 7\\ 7 &amp; 2 &amp; 2\\ 6 &amp; 3 &amp; 1
\end{array} \right| = 63$$

Find without expansion

\begin{description}
\item[(i)]
$\left|\begin{array} {ccc} 50 &amp; 4 &amp; 7\\ 70 &amp; 2 &amp; 2\\ 60 &amp; 3 &amp; 1
\end{array} \right|$

\item[(ii)]
$\left|\begin{array} {ccc} 15 &amp; 20 &amp; -56\\ 21 &amp; 10 &amp; -16\\ 18 &amp; 15
&amp; -8 \end{array} \right|$

\item[(iii)]
$\left|\begin{array} {ccc} 5x^3 &amp; \frac{2}{x} &amp; 21y^2\\ 7x^3 &amp;
\frac{1}{x} &amp; 6y^2\\ 6x^3 &amp; \frac{3}{2x} &amp; 3y^2
\end{array} \right|$

\end{description}

\medskip

\newpage
{\bf Answer}

$\left|\begin{array} {ccc} 5 &amp; 4 &amp; 7\\ 7 &amp; 2 &amp; 2\\ 6 &amp; 3 &amp; 1
\end{array} \right| = 63$

We will use that multiplication of 1 row/column by $k$ scales the
determinant value by $k$.

\begin{description}
\item[(i)]
\begin{eqnarray*} \left|\begin{array} {ccc} 50 &amp; 4 &amp; 7\\ 70 &amp; 2 &amp; 2\\ 60 &amp; 3 &amp;
1 \end{array} \right| &amp; = &amp; \left|\begin{array} {ccc} 10 \times 5
&amp; 4 &amp; 7\\ 10 \times 7 &amp; 2 &amp; 2\\ 10 \times 6 &amp; 3 &amp; 1 \end{array}
\right|\\ &amp; = &amp; 10\left|\begin{array} {ccc} 5 &amp; 4 &amp; 7\\ 7 &amp; 2 &amp;
2\\ 6 &amp; 3 &amp; 1 \end{array} \right|\\ &amp; = &amp; 10 \times 63\\ &amp; = &amp;
\un{630} \end{eqnarray*}

\item[(ii)]
\begin{eqnarray*} \left|\begin{array} {ccc} 15 &amp; 20 &amp; -56\\ 21 &amp; 10 &amp; -16\\ 18 &amp; 15 &amp;
-8 \end{array} \right| &amp; = &amp; \left|\begin{array} {ccc} 3 \times 5
&amp; 5 \times 4 &amp; -8 \times 2\\ 3 \times 6 &amp; 5 \times 3 &amp; -8 \times 1
\\ 3 \times 6 &amp; 5 \times 3 &amp; -8 \times 1 \end{array} \right|\\
&amp; = &amp; 3 \times \left|\begin{array} {ccc} 5 &amp; 5 \times 4 &amp; -8
\times 7\\ 7 &amp; 5 \times 2 &amp; -8 \times 2\\ 6 &amp; 5 \times 3 &amp; -8
\times 1 \end{array} \right|\\ &amp; = &amp; 3 \times 5 \times
\left|\begin{array} {ccc} 5 &amp; 4 &amp; -8 \times 7\\ 7 &amp; 2 &amp; -8 \times
2\\ 6 &amp; 3 &amp; -8\times 1 \end{array} \right|\\ &amp; = &amp; 3 \times 5
\times -8 \times \left|\begin{array} {ccc} 5 &amp; 4 &amp; 7\\ 7 &amp; 2 &amp; 2\\
6 &amp; 3 &amp; 1 \end{array} \right|\\ &amp; = &amp; 3 \times 5 \times -8 \times
63\\ &amp; = &amp; \un{-7560} \end{eqnarray*}

\newpage
\item[(iii)]
\begin{eqnarray*} \left|\begin{array} {ccc}
5x^3 &amp; \frac{2}{x} &amp; 21y^2\\ 7x^3 &amp; \frac{1}{x} &amp; 6y^2\\ 6x^3 &amp;
\frac{3}{2x} &amp; 3y^2 \end{array} \right| &amp; = &amp; x^3
\left|\begin{array} {ccc} 5 &amp; \frac{2}{x} &amp; 21y^2\\ 7 &amp;
\frac{1}{x} &amp; 6y^2\\ 6 &amp; \frac{3}{2x} &amp; 3y^2\end{array} \right|\\
&amp; = &amp; x^3 \times \ds\frac{1}{x} \times \left|\begin{array} {ccc} 5
&amp; 2 &amp; 21y^2\\ 7 &amp; 1 &amp; 6y^2\\ 6 &amp; \frac{3}{2} &amp; 3y^2
\end{array} \right|\\ &amp; = &amp; x^3 \times \ds\frac{1}{x} \times y^2 \times \left|\begin{array}
{ccc} 5 &amp; 2 &amp; 21\\ 7 &amp; 1 &amp; 6\\ 6 &amp; \frac{3}{2} &amp; 3
\end{array} \right|\\ x^2y^2 \left|\begin{array}
{ccc} 5 &amp; \frac{1}{2} \times 4 &amp; 21\\ 7 &amp; \frac{1}{2}\times 2 &amp;
6\\ 6 &amp; \frac{1}{2}\times 3 &amp; 3 \end{array} \right|\\ &amp; = &amp;
x^2y^2\times \ds\frac{1}{2} \left|\begin{array} {ccc} 5 &amp;  4 &amp; 3
\times 7\\ 7 &amp; 2 &amp; 3 \times 2\\ 6 &amp; 3 &amp; 3 \times 1 \end{array}
\right|\\ &amp; = &amp; \ds\frac{3}{2}x^2y^2 \left|\begin{array} {ccc} 5 &amp;
4 &amp; 7\\ 7 &amp; 2 &amp; 2\\ 6 &amp; 3 &amp; 1 \end{array} \right|\\ &amp; = &amp;
\ds\frac{3}{2}x^2y^2 \times 63\\ &amp; = &amp; \un{\ds\frac{189}{2}x^2y^2}
\end{eqnarray*}

\end{description}

\end{document}
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