\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\parindent=0pt
\begin{document}

{\bf Question}

Given that

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

Find without expansion

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

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

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

\end{description}

\medskip

\newpage
{\bf Answer}

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

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 & 4 & 7\\ 70 & 2 & 2\\ 60 & 3 &
1 \end{array} \right| & = & \left|\begin{array} {ccc} 10 \times 5
& 4 & 7\\ 10 \times 7 & 2 & 2\\ 10 \times 6 & 3 & 1 \end{array}
\right|\\ & = & 10\left|\begin{array} {ccc} 5 & 4 & 7\\ 7 & 2 &
2\\ 6 & 3 & 1 \end{array} \right|\\ & = & 10 \times 63\\ & = &
\un{630} \end{eqnarray*}

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

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

\end{description}

\end{document}
