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.