Question

Find the value of \[\left| \begin{gathered}
  & 53\,\,\,\,\,\,\,\,\,\,106\,\,\,\,\,\,\,\,\,159 \\
 & 52\,\,\,\,\,\,\,\,\,\,65\,\,\,\,\,\,\,\,\,\,\,\,91 \\
 & 102\,\,\,\,\,\,\,153\,\,\,\,\,\,\,\,\,\,\,221 \\
\end{gathered} \right|\]

Answer 1

Hint: We will make use of the determinant concept of expansion to find the answer for this sum. the expansion theory is;
If a determinant \[\left|\begin{gathered}
  & a\,\,\,\,\,\,\,\,\,\,b\,\,\,\,\,\,\,\,\,c \\
 & d\,\,\,\,\,\,\,\,\,e\,\,\,\,\,\,\,\,\,\,f \\
 & g\,\,\,\,\,\,\,\,\,h\,\,\,\,\,\,\,\,\,\,\,i \\
\end {gathered} \right|\] Is given then the value of this determinant can be found out by expanding it.also we must take note of the sign convention of a determinant which is.
\[|  & +\,\,\,\,\,\,\,\,\,-\,\,\,\,\,\,\,\,\,+ \\
 & -\,\,\,\,\,\,\,\,\,+\,\,\,\,\,\,\,\,\,- \\
 & +\,\,\,\,\,\,\,\,\,-\,\,\,\,\,\,\,\,\,+ \\ |\]
To expand a determinant;
\[
  & \left| \begin{gathered}  
  & a\,\,\,\,\,\,\,\,\,\,b\,\,\,\,\,\,\,\,\,c \\
 & d\,\,\,\,\,\,\,\,\,e\,\,\,\,\,\,\,\,\,\,f \\
 & g\,\,\,\,\,\,\,\,\,h\,\,\,\,\,\,\,\,\,\,\,i \\
\end {gathered} \right| \]
 & \left( a\times \left| \begin{gathered}  
  & e\,\,\,\,\,\,\,f \\
 & h\,\,\,\,\,\,\,\,i \\
\end {gathered} \right|\] \[ \right)-\left( b\times \left| \begin {gathered}  
  & d\,\,\,\,\,\,\,f \\
 & g\,\,\,\,\,\,\,\,i \\
\end {gathered} \right| \]  \[ \right)+\left( c\times \left| \begin {gathered}  
  & d\,\,\,\,\,\,\,e \\
 & g\,\,\,\,\,\,\,\,h \\
\end {gathered} \right| \right) \]

\[ \left[ a((e\times i)-(f\times h)) \right]-\left[ b((d\times i)-(f\times g)) \right]+\left[ c((d\times h)-(e\times g)) \right] \]

Complete step-by-step answer:
Using the expansion theory of determinant:
\[
  & \left|  \begin{gathered}  
  & 53\,\,\,\,\,\,\,\,\,\,106\,\,\,\,\,\,\,\,\,159 \\
 & 52\,\,\,\,\,\,\,\,\,\,65\,\,\,\,\,\,\,\,\,\,\,\,91 \\
 & 102\,\,\,\,\,\,\,153\,\,\,\,\,\,\,\,\,\,\,221 \\
\end {gathered} \right| \\
 & \\
 & \left( 53\times \left| \begin{gathered}  
  & 65\,\,\,\,\,\,\,\,\,\,\,\,\,91 \\
 & 153\,\,\,\,\,\,\,\,221 \\
\end {gathered} \right| \right)-\left( 106\times \left| \begin{gathered}   
  & 52\,\,\,\,\,\,\,\,\,\,\,91 \\
 & 102\,\,\,\,\,\,221 \\
\end {gathered} \right| \right)+\left( 159\times \left| \begin{gathered}   
  & 52\,\,\,\,\,\,\,\,\,\,\,\,\,65 \\
 & 102\,\,\,\,\,\,\,\,153 \\
\end {gathered} \right| \right) \\
 & \left[ 53((65\times 221)-(91\times 153)) \right]-\left[ 106((52\times 221)-(91\times 102)) \right]+\left[ 159((52\times 153)-(65\times 102)) \right] \\
 & \left[ 53(14365-13923) \right]-\left[ 106(11492-9282) \right]+\left[ 159(7956-6630) \right] \\
 & 23426-234260+210834=0 \\
\]
The value of this determinant is 0.

Note: The sign convention was taken into note while expanding the determinant thus the negative sign comes in the middle term. Observe the brackets while expanding carefully.