Concept:
Determinant of Matrix \(M = {\left[ {\begin{array}{*{20}{c}}
a\\
b\\
:\\
d
\end{array}} \right]_{n \times 1}}{\left[ {p\;\;\;q\;\;\;..\;\;s} \right]_{1 \times n\;}}\) is 0
Calculation:
\(A = {\left[ {\begin{array}{*{20}{c}}
2\\
{ - 4}\\
7
\end{array}} \right]_{3 \times 1}}{\left[ {1\;\;\;9\;\;\;5} \right]_{1 \times 3\;}}\;\)
\(A = \;{\left[ {\begin{array}{*{20}{c}}
2&{18}&{10}\\
{ - 4}&{ - 36}&{ - 20}\\
7&{63}&{35}
\end{array}} \right]_{3 \times 3}}\)
\(\left| A \right| = \;\left| {\begin{array}{*{20}{c}}
2&{18}&{10}\\
{ - 4}&{ - 36}&{ - 20}\\
7&{63}&{35}
\end{array}} \right|\)
\(\left| A \right| = \;2 \times - 4 \times 7\left| {\begin{array}{*{20}{c}}
1&9&5\\
1&9&5\\
1&9&5
\end{array}} \right|\)
Since 2 rows are identical determinant is 0
\(\left| A \right| = \;2 \times - 4 \times 7 \times 0 = 0\)
The determinant of A is equal to 0
