Question

Evaluate the modulus of the determinant: $\left|\begin{array}{ccc}18& 40& 89\\ 40& 89& 198\\ 89& 198& 440\end{array}\right|$

Answer 1

Applying $R_2-2R_1,R_3-5R_1$, we get
$△=$$\left|\begin{array}{ccc}18& 40& 89\\ 4& 9& 20\\ -1& -2& -5\end{array}\right|$
$△$ $\left|\begin{array}{ccc}18& 40& 9\\ 4& 9& 2\\ -1& -2& -1\end{array}\right|$
Now apply $C_1-2C_3$ to make zeros in $C_1$
$△$= $\left|\begin{array}{ccc}0& 40& 9\\ 0& 9& 2\\ 1& -2& -1\end{array}\right|$ = 1 $\left|\begin{array}{cc}40& 9\\ 9& 2\end{array}\right|$= 80 - 81 = -1.