Question

Using properties of determinants or otherwise evaluate $\left|\begin{array}{ccc}18& 40& 89\\ 40& 89& 198\\ 89& 198& 440\end{array}\right|$

• A. -1
• B. 0
• C. 9062
• D. None of these

Answer 1

The correct option is A -1

$\Delta =\left|\begin{array}{ccc}18& 40& 89\\ 40& 89& 198\\ 89& 198& 440\end{array}\right|=\left|\begin{array}{ccc}20-1& 40& 80+9\\ 40& 80+9& 200-2\\ 80+9& 200-2& 400+40\end{array}\right|=\left|\begin{array}{ccc}20& 40& 80\\ 40& 80& 200\\ 80& 200& 400\end{array}\right|+\left|\begin{array}{ccc}-1& 0& 9\\ 0& 9& -2\\ 9& -2& 40\end{array}\right|=\left(20\right)\left(40\right)\left(40\right)\left|\begin{array}{ccc}1& 2& 2\\ 2& 2& 5\\ 4& 5& 10\end{array}\right|+\left|\begin{array}{ccc}-1& 0& 9\\ 0& 9& -2\\ 9& -2& 40\end{array}\right|=\left(3200\right)(1(20-25)-2(20-20)+2(10-8))+-1(360-4)-0+9(0-81)=-1$